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timestamp[us]date 1999-12-11 03:00:00
2025-04-25 01:21:50
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A000001 | [
"M0098",
"N0035"
] | 294 | 2025-02-21T13:07:04 | [
"0",
"1",
"1",
"1",
"2",
"1",
"2",
"1",
"5",
"2",
"2",
"1",
"5",
"1",
"2",
"1",
"14",
"1",
"5",
"1",
"5",
"2",
"2",
"1",
"15",
"2",
"2",
"5",
"4",
"1",
"4",
"1",
"51",
"1",
"2",
"1",
"14",
"1",
"2",
"2",
"14",
"1",
"6",
"1",
"4",
"2",
"2",
"1",
"52",
"2",
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"2",
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"2",
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"1",
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"4",
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"1",
"4",
"1",
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"1",
"4",
"1",
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"1",
"2",
"3",
"4",
"1",
"6",
"1",
"52",
"15",
"2",
"1",
"15",
"1",
"2",
"1",
"12",
"1",
"10",
"1",
"4",
"2"
] | Number of groups of order n. | [
"A000001",
"A000019",
"A000637",
"A000638",
"A000679",
"A000688",
"A001034",
"A001228",
"A002106",
"A003277",
"A005180",
"A005432",
"A023675",
"A023676",
"A027623",
"A046057",
"A046058",
"A046059",
"A051532",
"A060689",
"A063756"
] | N. J. A. Sloane | [
"nonn",
"core",
"nice",
"hard"
] | 0 | 5 | oeisdata/seq/A000/A000001.seq | %I A000001 M0098 N0035 #294 Feb 21 2025 13:07:04
%S A000001 0,1,1,1,2,1,2,1,5,2,2,1,5,1,2,1,14,1,5,1,5,2,2,1,15,2,2,5,4,1,4,1,51,
%T A000001 1,2,1,14,1,2,2,14,1,6,1,4,2,2,1,52,2,5,1,5,1,15,2,13,2,2,1,13,1,2,4,
%U A000001 267,1,4,1,5,1,4,1,50,1,2,3,4,1,6,1,52,15,2,1,15,1,2,1,12,1,10,1,4,2
%N A000001 Number of groups of order n.
%C A000001 Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - _Lekraj Beedassy_, Dec 16 2004
%C A000001 Also, number of nonisomorphic primitives of the combinatorial species Lin[n-1]. - _Nicolae Boicu_, Apr 29 2011
%C A000001 In (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), a(n) is called the "group number of n", denoted by gnu(n), and the first occurrence of k is called the "minimal order attaining k", denoted by moa(k) (see A046057). - _Daniel Forgues_, Feb 15 2017
%C A000001 It is conjectured in (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008) that the sequence n -> a(n) -> a(a(n)) = a^2(n) -> a(a(a(n))) = a^3(n) -> ... -> consists ultimately of 1s, where a(n), denoted by gnu(n), is called the "group number of n". - _Muniru A Asiru_, Nov 19 2017
%C A000001 MacHale (2020) shows that there are infinitely many values of n for which there are more groups than rings of that order (cf. A027623). He gives n = 36355 as an example. It would be nice to have enough values of n to create an OEIS entry for them. - _N. J. A. Sloane_, Jan 02 2021
%C A000001 I conjecture that a(i) * a(j) <= a(i*j) for all nonnegative integers i and j. - _Jorge R. F. F. Lopes_, Apr 21 2024
%D A000001 S. R. Blackburn, P. M. Neumann, and G. Venkataraman, Enumeration of Finite Groups, Cambridge, 2007.
%D A000001 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 302, #35.
%D A000001 J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209.
%D A000001 H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
%D A000001 CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 150.
%D A000001 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley Publ. Co., Reading, MA, 1989, Section 6.6 'Fibonacci Numbers' pp. 281-283.
%D A000001 M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
%D A000001 D. Joyner, 'Adventures in Group Theory', Johns Hopkins Press. Pp. 169-172 has table of groups of orders < 26.
%D A000001 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.
%D A000001 M. F. Newman and E. A. O'Brien, A CAYLEY library for the groups of order dividing 128. Group theory (Singapore, 1987), 437-442, de Gruyter, Berlin-New York, 1989.
%D A000001 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000001 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000001 H.-U. Besche and Ivan Panchenko, <a href="/A000001/b000001.txt">Table of n, a(n) for n = 0..2047</a> [Terms 1 through 2015 copied from Small Groups Library mentioned below. Terms 2016 - 2047 added by _Ivan Panchenko_, Aug 29 2009. 0 prepended by _Ray Chandler_, Sep 16 2015. a(1024) corrected by _Benjamin Przybocki_, Jan 06 2022]
%H A000001 H. A. Bender, <a href="http://www.jstor.org/stable/1967981">A determination of the groups of order p^5</a>, Ann. of Math. (2) 29, pp. 61-72 (1927).
%H A000001 Hans Ulrich Besche and Bettina Eick, <a href="http://dx.doi.org/10.1006/jsco.1998.0258">Construction of finite groups</a>, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
%H A000001 Hans Ulrich Besche and Bettina Eick, <a href="http://dx.doi.org/10.1006/jsco.1998.0259">The groups of order at most 1000 except 512 and 768</a>, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
%H A000001 H. U. Besche, B. Eick and E. A. O'Brien, <a href="http://www.ams.org/era/2001-07-01/S1079-6762-01-00087-7/home.html">The groups of order at most 2000</a>, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.
%H A000001 H. U. Besche, B. Eick, E. A. O'Brien and Max Horn, <a href="https://gap-packages.github.io/smallgrp/">The Small Groups Library</a>
%H A000001 H. U. Besche, B. Eick and E. A. O'Brien, <a href="https://web.archive.org/web/20161030100727/http://www.icm.tu-bs.de/ag_algebra/software/small/number.html">Number of isomorphism types of finite groups of given order</a> [gives incorrect a(1024)]
%H A000001 H.-U. Besche, B. Eick and E. A. O'Brien, <a href="http://dx.doi.org/10.1142/S0218196702001115">A Millennium Project: Constructing Small Groups</a>, Internat. J. Algebra and Computation, 12 (2002), 623-644.
%H A000001 Henry Bottomley, <a href="/A000001/a000001.gif">Illustration of initial terms</a>
%H A000001 David Burrell, <a href="https://doi.org/10.1080/00927872.2021.2006680">On the number of groups of order 1024</a>, Communications in Algebra, 2021, 1-3.
%H A000001 J. H. Conway, Heiko Dietrich and E. A. O'Brien, <a href="http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf">Counting groups: gnus, moas and other exotica</a>, Math. Intell., Vol. 30, No. 2, Spring 2008.
%H A000001 Marek Dančo, Mikoláš Janota, Michael Codish, and João Jorge Araújo, <a href="https://arxiv.org/abs/2502.10155">Complete Symmetry Breaking for Finite Models</a>, arXiv:2502.10155 [cs.LO], 2025. See p. 7. Mentions this sequence.
%H A000001 Yang-Hui He and Minhyong Kim, <a href="https://arxiv.org/abs/1905.02263">Learning Algebraic Structures: Preliminary Investigations</a>, arXiv:1905.02263 [cs.LG], 2019.
%H A000001 Otto Hölder, <a href="http://dx.doi.org/10.1007/BF01443651">Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4</a>, Math. Ann. 43 pp. 301-412 (1893).
%H A000001 Max Horn, <a href="https://groups.quendi.de/">Numbers of isomorphism types of finite groups of given order</a>
%H A000001 Rodney James, <a href="http://dx.doi.org/10.1090/S0025-5718-1980-0559207-0">The groups of order p^6 (p an odd prime)</a>, Math. Comp. 34 (1980), 613-637.
%H A000001 Rodney James and John Cannon, <a href="http://dx.doi.org/10.1090/S0025-5718-1969-0238953-8">Computation of isomorphism classes of p-groups</a>, Mathematics of Computation 23.105 (1969): 135-140.
%H A000001 Olexandr Konovalov, <a href="https://github.com/olexandr-konovalov/gnu/tree/master">Crowdsourcing project for the database of numbers of isomorphism types of finite groups</a>, Github (a list of gnu(n) for many n < 50000).
%H A000001 Desmond MacHale, <a href="https://doi.org/10.1080/00029890.2020.1820790">Are There More Finite Rings than Finite Groups?</a>, Amer. Math. Monthly (2020) Vol. 127, Issue 10, 936-938.
%H A000001 Mehdi Makhul, Josef Schicho, and Audie Warren, <a href="https://arxiv.org/abs/2306.04392">On Galois groups of type-1 minimally rigid graphs</a>, arXiv:2306.04392 [math.CO], 2023.
%H A000001 G. A. Miller, <a href="http://www.jstor.org/stable/2370630">Determination of all the groups of order 64</a>, Amer. J. Math., 52 (1930), 617-634.
%H A000001 László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
%H A000001 Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/MAA/07-Sequence%20Pictures/mathgames_12_08_03.html">Sequence Pictures</a>, Math Games column, Dec 08 2003.
%H A000001 Ed Pegg, Jr., <a href="/A000043/a000043_2.pdf">Sequence Pictures</a>, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
%H A000001 D. S. Rajan, <a href="http://dx.doi.org/10.1016/0012-365X(93)90061-W">The equations D^kY=X^n in combinatorial species</a>, Discrete Mathematics 118 (1993) 197-206 North-Holland.
%H A000001 E. Rodemich, <a href="http://dx.doi.org/10.1016/0021-8693(90)90244-I">The groups of order 128</a>, J. Algebra 67 (1980), no. 1, 129-142.
%H A000001 Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/data.html">Combinatorial Catalogues</a>. See subpage "Generators of small groups" for explicit generators for most groups of even order < 1000. [broken link]
%H A000001 D. Rusin, <a href="/A000001/a000001.txt">Asymptotics</a> [Cached copy of lost web page]
%H A000001 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FiniteGroup.html">Finite Group</a>
%H A000001 Wikipedia, <a href="http://en.wikipedia.org/wiki/Finite_group">Finite group</a>
%H A000001 M. Wild, <a href="http://www.jstor.org/stable/30037381">The groups of order sixteen made easy</a>, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
%H A000001 Gang Xiao, <a href="http://wims.unice.fr/~wims/wims.cgi?module=tool/algebra/smallgroup">SmallGroup</a>
%H A000001 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H A000001 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F A000001 From _Mitch Harris_, Oct 25 2006: (Start)
%F A000001 For p, q, r primes:
%F A000001 a(p) = 1, a(p^2) = 2, a(p^3) = 5, a(p^4) = 14, if p = 2, otherwise 15.
%F A000001 a(p^5) = 61 + 2*p + 2*gcd(p-1,3) + gcd(p-1,4), p >= 5, a(2^5)=51, a(3^5)=67.
%F A000001 a(p^e) ~ p^((2/27)e^3 + O(e^(8/3))).
%F A000001 a(p*q) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p. (p < q)
%F A000001 a(p*q^2) is one of the following:
%F A000001 ---------------------------------------------------------------------------
%F A000001 | a(p*q^2) | p*q^2 of the form | Sequences (p*q^2) |
%F A000001 ---------- ------------------------------------------ ---------------------
%F A000001 | (p+9)/2 | q == 1 (mod p), p odd | A350638 |
%F A000001 | 5 | p=3, q=2 => p*q^2 = 12 |Special case with A_4|
%F A000001 | 5 | p=2, q odd | A143928 |
%F A000001 | 5 | p == 1 (mod q^2) | A350115 |
%F A000001 | 4 | p == 1 (mod q), p > 3, p !== 1 (mod q^2) | A349495 |
%F A000001 | 3 | q == -1 (mod p), p and q odd | A350245 |
%F A000001 | 2 | q !== +-1 (mod p) and p !== 1 (mod q) | A350422 |
%F A000001 ---------------------------------------------------------------------------
%F A000001 [Table from _Bernard Schott_, Jan 18 2022]
%F A000001 a(p*q*r) (p < q < r) is one of the following:
%F A000001 q == 1 (mod p) r == 1 (mod p) r == 1 (mod q) a(p*q*r)
%F A000001 -------------- -------------- -------------- --------
%F A000001 No No No 1
%F A000001 No No Yes 2
%F A000001 No Yes No 2
%F A000001 No Yes Yes 4
%F A000001 Yes No No 2
%F A000001 Yes No Yes 3
%F A000001 Yes Yes No p+2
%F A000001 Yes Yes Yes p+4
%F A000001 [table from Derek Holt].
%F A000001 (End)
%F A000001 a(n) = A000688(n) + A060689(n). - _R. J. Mathar_, Mar 14 2015
%e A000001 Groups of orders 1 through 10 (C_n = cyclic, D_n = dihedral of order n, Q_8 = quaternion, S_n = symmetric):
%e A000001 1: C_1
%e A000001 2: C_2
%e A000001 3: C_3
%e A000001 4: C_4, C_2 X C_2
%e A000001 5: C_5
%e A000001 6: C_6, S_3=D_6
%e A000001 7: C_7
%e A000001 8: C_8, C_4 X C_2, C_2 X C_2 X C_2, D_8, Q_8
%e A000001 9: C_9, C_3 X C_3
%e A000001 10: C_10, D_10
%p A000001 GroupTheory:-NumGroups(n); # with(GroupTheory); loads this command - _N. J. A. Sloane_, Dec 28 2017
%t A000001 FiniteGroupCount[Range[100]] (* _Harvey P. Dale_, Jan 29 2013 *)
%t A000001 a[ n_] := If[ n < 1, 0, FiniteGroupCount @ n]; (* _Michael Somos_, May 28 2014 *)
%o A000001 (Magma) D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] ]; // _John Cannon_, Dec 23 2006
%o A000001 (GAP) A000001 := Concatenation([0], List([1..500], n -> NumberSmallGroups(n))); # _Muniru A Asiru_, Oct 15 2017
%Y A000001 The main sequences concerned with group theory are A000001 (this one), A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A000688, A060689, A051532.
%Y A000001 Cf. A046058, A046059, A023675, A023676.
%Y A000001 A003277 gives n for which A000001(n) = 1, A063756 (partial sums).
%Y A000001 A046057 gives first occurrence of each k.
%Y A000001 A027623 gives the number of rings of order n.
%K A000001 nonn,core,nice,hard
%O A000001 0,5
%A A000001 _N. J. A. Sloane_
%E A000001 More terms from _Michael Somos_
%E A000001 Typo in b-file description fixed by _David Applegate_, Sep 05 2009
| eeb417502ac5f2c5dd40a6442b27eb51 |
A000002 | [
"M0190",
"N0070"
] | 390 | 2025-02-16T08:32:18 | [
"1",
"2",
"2",
"1",
"1",
"2",
"1",
"2",
"2",
"1",
"2",
"2",
"1",
"1",
"2",
"1",
"1",
"2",
"2",
"1",
"2",
"1",
"1",
"2",
"1",
"2",
"2",
"1",
"1",
"2",
"1",
"1",
"2",
"1",
"2",
"2",
"1",
"2",
"2",
"1",
"1",
"2",
"1",
"2",
"2",
"1",
"2",
"1",
"1",
"2",
"1",
"1",
"2",
"2",
"1",
"2",
"2",
"1",
"1",
"2",
"1",
"2",
"2",
"1",
"2",
"2",
"1",
"1",
"2",
"1",
"1",
"2",
"1",
"2",
"2",
"1",
"2",
"1",
"1",
"2",
"2",
"1",
"2",
"2",
"1",
"1",
"2",
"1",
"2",
"2",
"1",
"2",
"2",
"1",
"1",
"2",
"1",
"1",
"2",
"2",
"1",
"2",
"1",
"1",
"2",
"1",
"2",
"2"
] | Kolakoski sequence: a(n) is length of n-th run; a(1) = 1; sequence consists just of 1's and 2's. | [
"A000002",
"A001083",
"A001462",
"A005041",
"A006928",
"A010060",
"A013947",
"A042942",
"A049705",
"A054353",
"A054354",
"A064353",
"A069864",
"A071820",
"A071907",
"A071928",
"A071942",
"A074286",
"A074803",
"A074804",
"A078880",
"A078929",
"A079729",
"A079730",
"A088568",
"A088569",
"A100144",
"A100428",
"A100429",
"A118270",
"A156077",
"A171899",
"A216345",
"A234322",
"A294447"
] | N. J. A. Sloane | [
"nonn",
"core",
"easy",
"nice"
] | 0 | 5 | oeisdata/seq/A000/A000002.seq | %I A000002 M0190 N0070 #390 Feb 16 2025 08:32:18
%S A000002 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,
%T A000002 2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,
%U A000002 2,1,1,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2
%N A000002 Kolakoski sequence: a(n) is length of n-th run; a(1) = 1; sequence consists just of 1's and 2's.
%C A000002 Historical note: the sequence might be better called the Oldenburger-Kolakoski sequence, since it was discussed by Rufus Oldenburger in 1939; see links. - _Clark Kimberling_, Dec 06 2012. However, to avoid confusion, this sequence will be known in the OEIS as the Kolakoski sequence. It is undesirable to have some entries refer to the Oldenburger-Kolakoski sequence and others to the Kolakoski sequence. - _N. J. A. Sloane_, Nov 22 2017
%C A000002 It is an unsolved problem to show that the density of 1's is equal to 1/2.
%C A000002 A weaker problem is to construct a combinatorial bijection between the set of positions of 1's and the set of positions of 2's. - _Gus Wiseman_, Mar 01 2016
%C A000002 The sequence is cubefree and all square subwords have lengths which are one of 2, 4, 6, 18 and 54 (see A294447) [Carpi, 1994].
%C A000002 This is a fractal sequence: replace each run with its length and recover the original sequence. - _Kerry Mitchell_, Dec 08 2005
%C A000002 Kupin and Rowland write: We use a method of Goulden and Jackson to bound freq_1(K), the limiting frequency of 1 in the Kolakoski word K. We prove that |freq_1(K) - 1/2| <= 17/762, assuming the limit exists and establish the semirigorous bound |freq_1(K) - 1/2| <= 1/46. - _Jonathan Vos Post_, Sep 16 2008
%C A000002 freq_1(K) is conjectured to be 1/2 + O(log(K)) (see PlanetMath link). - _Jon Perry_, Oct 29 2014
%C A000002 Conjecture: Taking the sequence in word lengths of 10, for example, batch 1-10, 11-20, etc., then there can only be 4, 5 or 6 1's in each batch. - _Jon Perry_, Sep 26 2012
%C A000002 From _Jean-Christophe Hervé_, Oct 04 2014: (Start)
%C A000002 The sequence does not contain words of the form ababa, because this would imply the impossible 111 (1 b, 1 a, 1 b) somewhere before. This demonstrates the conjecture made by _Jon Perry_: more than 6 1's or 6 2's in a word of 10 would necessitate something like aabaabaaba, which would imply the impossible 12121 before (word aabaababaa is also impossible because of ababa). The remark on the sextuplets below even shows that the number of 1's in any 9-tuplet is always 4 or 5.
%C A000002 There are only 6 triples that appear in the sequence (112, 121, 122, 211, 212 and 221); and by the preceding argument, only 18 sextuplets: the 6 double triples (112112, etc.); 112122, 112212, 121122, 121221, 211212, and 211221; and those obtained by reversing the order of the triples (122112, etc.). Regarding the density of 1's in the sequence, these 12 sextuplets all have a density 1/2 of 1's, and the 6 double triples all lead to a word with this exact density after transformation by the Kolakoski rules, for example: 112112 -> 12112122 (4 1's/8); this is because the second triple reverses the numbers of 1's and 2's generated by the first triple. Therefore, the sequence can be split into the double triples on one side, a part whose transformation (which is in the sequence) has a density of 1's of 1/2; and a part with the other sextuplets, which has directly the same density of 1's. (End)
%C A000002 If we map 1 to +1 and 2 to -1, then the mapped sequence would have a [conjectured] mean of 0, since the Kolakoski sequence is [conjectured] to have an equal density (1/2) of 1s and 2s. For the partial sums of this mapped sequence, see A088568. - _Daniel Forgues_, Jul 08 2015
%C A000002 Looking at the plot for A088568, it seems that although the asymptotic densities of 1s and 2s appear to be 1/2, there might be a bias in favor of the 2s. I.e., D(1) = 1/2 - O(log(n)/n), D(2) = 1/2 + O(log(n)/n). - _Daniel Forgues_, Jul 11 2015
%C A000002 From _Michel Dekking_, Jan 31 2018: (Start)
%C A000002 (a(n)) is the unique fixed point of the 2-block substitution beta
%C A000002 11 -> 12
%C A000002 12 -> 122
%C A000002 21 -> 112
%C A000002 22 -> 1122.
%C A000002 A 2-block substitution beta maps a word w(1)...w(2n) to the word
%C A000002 beta(w(1)w(2))...beta(w(2n-1)w(2n)).
%C A000002 If the word has odd length, then the last letter is ignored.
%C A000002 It was noted by me in 1979 in the Bordeaux seminar on number theory that (a(n+1)) is fixed point of the 2-block substitution 11 -> 21, 12 -> 211, 21 -> 221, 22 -> 2211. (End)
%C A000002 Named after the American artist and recreational mathematician William George Kolakoski (1944-1997). - _Amiram Eldar_, Jun 17 2021
%D A000002 Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 337.
%D A000002 Éric Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
%D A000002 F. M. Dekking, What Is the Long Range Order in the Kolakoski Sequence?, in The mathematics of long-range aperiodic order (Waterloo, ON, 1995), 115-125, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 489, Kluwer Acad. Publ., Dordrecht, 1997. Math. Rev. 98g:11022.
%D A000002 Michael S. Keane, Ergodic theory and subshifts of finite type, Chap. 2 of T. Bedford et al., eds., Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford, 1991, esp. p. 50.
%D A000002 J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
%D A000002 Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
%D A000002 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000002 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000002 Ilan Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 233.
%H A000002 N. J. A. Sloane, <a href="/A000002/b000002.txt">Table of n, a(n) for n = 1..10502</a>
%H A000002 Jean-Paul Allouche, Michael Baake, Julien Cassaigne and David Damanik, <a href="https://arxiv.org/abs/math/0106121">Palindrome complexity</a>, arXiv:math/0106121 [math.CO], 2001; <a href="https://doi.org/10.1016/S0304-3975(01)00212-2">Theoretical Computer Science</a>, Vol. 292 (2003), pp. 9-31.
%H A000002 Michael Baake and Bernd Sing, <a href="https://arxiv.org/abs/math/0206098">Kolakoski-(3,1) is a (deformed) model set</a>, arXiv:math/0206098 [math.MG], 2002-2003.
%H A000002 Alex Bellos and Brady Haran, <a href="https://www.youtube.com/watch?v=co5sOgZ3XcM">The Kolakoski Sequence</a>, Numberphile video (2017).
%H A000002 Olivier Bordellès and Benoit Cloitre, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Bordelles/bordelles7r.html">Bounds for the Kolakoski Sequence</a>, J. Integer Sequences, Vol. 14 (2011), Article 11.2.1.
%H A000002 Richard P. Brent, <a href="https://maths-people.anu.edu.au/~brent/pd/Kolakoski-UNSW.pdf">Fast algorithms for the Kolakoski sequence</a>, Slides from a talk, 2016.
%H A000002 Éric Brier, Rémi Géraud-Stewart, David Naccache, Alessandro Pacco, and Emanuele Troiani, <a href="https://arxiv.org/abs/2006.07246">The Look-and-Say The Biggest Sequence Eventually Cycles</a>, arXiv:2006.07246 [math.DS], 2020.
%H A000002 Arturo Carpi, <a href="https://doi.org/10.1016/0020-0190(94)00162-6">On repeated factors in C^infinity-words</a>, Information Processing Letters, Vol. 52 (1994), pp. 289-294.
%H A000002 Benoit Cloitre, <a href="/A000002/a000002.pdf">The Kolakoski transform of words</a>.
%H A000002 Benoit Cloitre, <a href="/A000002/a000002.png">Plot of walk on the first 60000 terms (step of unit length moving right with angle Pi/2 if 2 and left with angle -Pi/2 if 1 starting at (0,0))</a>.
%H A000002 F. M. Dekking, <a href="https://www.jstor.org/stable/44165352">Regularity and irregularity of sequences generated by automata</a>, Seminar on Number Theory, 1979-1980 (Talence, 1979-1980), Exp. No. 9, 10 pp., Univ. Bordeaux I, Talence, 1980.
%H A000002 F. M. Dekking, <a href="http://www.digizeitschriften.de/dms/resolveppn/?PPN=GDZPPN002544490">On the structure of self-generating sequences</a>, Seminar on Number Theory, 1980-1981 (Talence, 1980-1981), Exp. No. 31, 6 pp., Univ. Bordeaux I, Talence, 1981. Math. Rev. 83e:10075.
%H A000002 F. M. Dekking, <a href="http://citeseerx.ist.psu.edu/pdf/7af40df61b38208d1eccca350f4869b6f1a6a18f">What Is the Long Range Order in the Kolakoski Sequence?</a>, Report 95-100, Technische Universiteit Delft, 1995.
%H A000002 F. M. Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Dekking/dek25.html">The Thue-Morse Sequence in Base 3/2</a>, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
%H A000002 Jörg Endrullis, Dimitri Hendriks and Jan Willem Klop, <a href="http://math.colgate.edu/~integers/a6num/a6num.pdf">Degrees of Streams</a>, Integers, Vol. 11B (2011), A6.
%H A000002 David Eppstein, <a href="https://11011110.github.io/blog/2016/10/13/kolakoski-tree.html">The Kolakoski tree</a> and <a href="https://11011110.github.io/blog/2016/10/14/kolakoski-sequence-via.html">The Kolakoski sequence via bit tricks instead of recursion</a>.
%H A000002 Jean-Marc Fédou and Gabriele Fici, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Fici/fici.html">Some remarks on differentiable sequences and recursivity</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.2.
%H A000002 Abdallah Hammam, <a href="http://pubs.sciepub.com/tjant/4/3/1/">Some new Formulas for the Kolakoski Sequence A000002</a>, Turkish Journal of Analysis and Number Theory, Vol. 4, No. 3 (2016), pp. 54-59.
%H A000002 Mari Huova and Juhani Karhumäki, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Huova/huova2.html">On Unavoidability of k-abelian Squares in Pure Morphic Words</a>, Journal of Integer Sequences, Vol. 16 (2013), #13.2.9.
%H A000002 Clark Kimberling, Integer Sequences and Arrays, <a href="http://faculty.evansville.edu/ck6/integer/index.html">Illustration of the Kolakoski sequence</a>.
%H A000002 William Kolakoski, <a href="https://www.jstor.org/stable/2313883">Problem 5304</a>, Amer. Math. Monthly, Vol. 72, No. 8 (1965), p. 674; <a href="https://www.jstor.org/stable/2314839">Self Generating Runs</a>, Solution to Problem 5304 by Necdet Üçoluk, Vol. 73, No. 6 (1966), pp. 681-682.
%H A000002 Leonid V. Kovalev, <a href="https://web.archive.org/web/20130606153459 /http://calculus7.org/2012/04/14/kolakoski-sequence-ii/">Kolakoski sequence II</a>.
%H A000002 Elizabeth J. Kupin and Eric S. Rowland, <a href="https://arxiv.org/abs/0809.2776">Bounds on the frequency of 1 in the Kolakoski word</a>, arXiv:0809.2776 [math.CO], Sep 16, 2008.
%H A000002 Rabie A. Mahmoud, <a href="https://www.researchgate.net/publication/271910838">Hardware Implementation of Binary Kolakoski Sequence</a>, Research Gate, 2015.
%H A000002 Kerry Mitchell, <a href="http://kerrymitchellart.com/articles/Spirolateral-Type_Images_from_Integer_Sequences.pdf">Spirolateral-Type Images from Integer Sequences</a>, 2013.
%H A000002 Johan Nilsson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Nilsson/nilsson5.html">A Space Efficient Algorithm for the Calculation of the Digit Distribution in the Kolakoski Sequence</a>, J. Int. Seq., Vol. 15 (2012), Article 12.6.7; <a href="https://arxiv.org/abs/1110.4228">arXiv preprint</a>, arXiv:1110.4228 [math.CO], Oct 19, 2011.
%H A000002 J. Nilsson, <a href="https://doi.org/10.12693/APhysPolA.126.549">Letter Frequencies in the Kolakoski Sequence</a>, Acta Physica Polonica A, Vol. 126 (2014), pp. 549-552.
%H A000002 Rufus Oldenburger, <a href="https://doi.org/10.1090/S0002-9947-1939-0000352-9">Exponent trajectories in symbolic dynamics</a>, Trans. Amer. Math. Soc., Vol. 46 (1939), pp. 453-466.
%H A000002 Matthew Parker, <a href="https://oeis.org/A000002/a000002_50K.7z">The first 50K terms (7-Zip compressed file)</a>.
%H A000002 Gheorghe Păun and Arto Salomaa, <a href="https://www.jstor.org/stable/2975113">Self-reading sequences</a>, Amer. Math. Monthly, Vol. 103, No. 2 (1996), pp. 166-168.
%H A000002 Michael Rao, <a href="https://www.arthy.org/kola/kola.php">Trucs et bidules sur la séquence de Kolakoski</a>, 2012, in French.
%H A000002 A. Scolnicov, <a href="http://planetmath.org/kolakoskisequence">Kolakoski sequence</a>, PlanetMath.org.
%H A000002 Bobby Shen, <a href="https://arxiv.org/abs/1703.00180">A uniformness conjecture of the Kolakoski sequence, graph connectivity, and correlations</a>, arXiv:1703.00180 [math.CO], 2017.
%H A000002 Bernd Sing, <a href="https://emis.de/journals/INTEGERS/papers/a14num/a14num.Abstract.html">More Kolakoski Sequences</a>, INTEGERS, Vol. 11B (2011), A14.
%H A000002 N. J. A. Sloane, <a href="/A001149/a001149.pdf">Handwritten notes on Self-Generating Sequences, 1970</a>. (note that A1148 has now become A005282)
%H A000002 N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, <a href="https://vimeo.com/314786942">Part I</a>, <a href="https://vimeo.com/314790822">Part 2</a>, <a href="https://oeis.org/A320487/a320487.pdf">Slides</a>. (Mentions this sequence)
%H A000002 Bertran Steinsky, <a href="https://www.cs.uwaterloo.ca/journals/JIS/VOL9/Steinsky/steinsky5.html">A Recursive Formula for the Kolakoski Sequence A000002</a>, J. Integer Sequences, Vol. 9 (2006), Article 06.3.7.
%H A000002 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KolakoskiSequence.html">Kolakoski Sequence</a>
%H A000002 Wikipedia, <a href="https://en.wikipedia.org/wiki/Kolakoski_sequence">Kolakoski sequence</a>.
%H A000002 Gus Wiseman, <a href="https://imgur.com/MqRlpnm">Kolakoski fractal animation for n=40000</a>.
%H A000002 Ed Wynn, <a href="/A000002/a000002.spl.txt">Program to generate A000002 in Shakespeare Programming Language</a>.
%H A000002 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F A000002 These two formulas define completely the sequence: a(1)=1, a(2)=2, a(a(1) + a(2) + ... + a(k)) = (3 + (-1)^k)/2 and a(a(1) + a(2) + ... + a(k) + 1) = (3 - (-1)^k)/2. - _Benoit Cloitre_, Oct 06 2003
%F A000002 a(n+2)*a(n+1)*a(n)/2 = a(n+2) + a(n+1) + a(n) - 3 (this formula doesn't define the sequence, it is just a consequence of the definition). - _Benoit Cloitre_, Nov 17 2003
%F A000002 a(n+1) = 3 - a(n) + (a(n) - a(n-1))*(a(b(n)) - 1), where b(n) is the sequence A156253. - Jean-Marc Fedou and _Gabriele Fici_, Mar 18 2010
%F A000002 a(n) = (3 + (-1)^A156253(n))/2. - _Benoit Cloitre_, Sep 17 2013
%F A000002 Conjectures from _Boštjan Gec_, Oct 07 2024: (Start)
%F A000002 a(n)*(a(n-1) + a(n-2) - 3) + a(n-1)*a(n-2) + 7 = 3*a(n-1) + 3*a(n-2).
%F A000002 a(n)*(a(n-1) + a(n-2) - 3) = a(n-3)*(a(n-1) + a(n-2) - 3). (End)
%F A000002 Comment from _Kevin Ryde_, Oct 07 2024: The above formulas are true: The parts identify when terms are same or different and they hold for any sequence of 1's and 2's with run lengths 1 or 2.
%e A000002 Start with a(1) = 1. By definition of the sequence, this says that the first run has length 1, so it must be a single 1, and a(2) = 2. Thus, the second run (which starts with this 2) must have length 2, so the third term must be also be a(3) = 2, and the fourth term can't be a 2, so must be a(4) = 1. Since a(3) = 2, the third run must have length 2, so we deduce a(5) = 1, a(6) = 2, and so on. The correction I made was to change a(4) to a(5) and a(5) to a(6). - _Labos Elemer_, corrected by _Graeme McRae_
%p A000002 M := 100; s := [ 1,2,2 ]; for n from 3 to M do for i from 1 to s[ n ] do s := [ op(s),1+((n-1)mod 2) ]; od: od: s; A000002 := n->s[n];
%p A000002 # alternative implementation based on the Cloitre formula:
%p A000002 A000002 := proc(n)
%p A000002 local ksu,k ;
%p A000002 option remember;
%p A000002 if n = 1 then
%p A000002 1;
%p A000002 elif n <=3 then
%p A000002 2;
%p A000002 else
%p A000002 for k from 1 do
%p A000002 ksu := add(procname(i),i=1..k) ;
%p A000002 if n = ksu then
%p A000002 return (3+(-1)^k)/2 ;
%p A000002 elif n = ksu+ 1 then
%p A000002 return (3-(-1)^k)/2 ;
%p A000002 end if;
%p A000002 end do:
%p A000002 end if;
%p A000002 end proc: # _R. J. Mathar_, Nov 15 2014
%t A000002 a[steps_] := Module[{a = {1, 2, 2}}, Do[a = Append[a, 1 + Mod[(n - 1), 2]], {n, 3, steps}, {i, a[[n]]}]; a]
%t A000002 a[ n_] := If[ n < 3, Max[ 0, n], Module[ {an = {1, 2, 2}, m = 3}, While[ Length[ an] < n, an = Join[ an, Table[ Mod[m, 2, 1], { an[[ m]]} ]]; m++]; an[[n]]]] (* _Michael Somos_, Jul 11 2011 *)
%t A000002 n=8; Prepend[ Nest[ Flatten[ Partition[#, 2] /. {{2, 2} -> {2, 2, 1, 1}, {2, 1} -> {2, 2, 1}, {1, 2} -> {2, 1, 1}, {1, 1} -> {2, 1}}] &, {2, 2}, n], 1] (* _Birkas Gyorgy_, Jul 10 2012 *)
%t A000002 KolakoskiSeq[n_Integer] := Block[{a = {1, 2, 2}}, Fold[Join[#1, ConstantArray[Mod[#2, 2] + 1, #1[[#2]]]] &, a, Range[3, n]]]; KolakoskiSeq[999] (* _Mikk Heidemaa_, Nov 01 2024 *)
%o A000002 (PARI) my(a=[1,2,2]); for(n=3,80, for(i=1,a[n],a=concat(a,2-n%2))); a
%o A000002 (PARI) {a(n) = local(an=[1, 2, 2], m=3); if( n<1, 0, while( #an < n, an = concat( an, vector(an[m], i, 2-m%2)); m++); an[n])};
%o A000002 (Haskell) a = 1:2: drop 2 (concat . zipWith replicate a . cycle $ [1,2]) -- _John Tromp_, Apr 09 2011
%o A000002 (Python)
%o A000002 # For explanation see link.
%o A000002 def Kolakoski():
%o A000002 x = y = -1
%o A000002 while True:
%o A000002 yield [2,1][x&1]
%o A000002 f = y &~ (y+1)
%o A000002 x ^= f
%o A000002 y = (y+1) | (f & (x>>1))
%o A000002 K = Kolakoski()
%o A000002 print([next(K) for _ in range(100)]) # _David Eppstein_, Oct 15 2016
%Y A000002 Cf. A001083, A006928, A042942, A069864, A010060, A078929, A171899, A054353 (partial sums), A074286, A216345, A294447.
%Y A000002 Cf. A054354, bisections: A100428, A100429.
%Y A000002 Cf. A013947, A156077, A234322 (positions, running total and percentage of 1's).
%Y A000002 Cf. A118270.
%Y A000002 Cf. A049705, A088569 (are either subsequences of A000002? - _Jon Perry_, Oct 30 2014)
%Y A000002 Kolakoski-type sequences using other seeds than (1,2):
%Y A000002 A078880 (2,1), A064353 (1,3), A071820 (2,3), A074804 (3,2), A071907 (1,4), A071928 (2,4), A071942 (3,4), A074803 (4,2), A079729 (1,2,3), A079730 (1,2,3,4).
%Y A000002 Other self-describing: A001462 (Golomb sequence, see also references therein), A005041, A100144.
%Y A000002 Cf. A088568 (partial sums of [3 - 2 * a(n)]).
%K A000002 nonn,core,easy,nice
%O A000002 1,2
%A A000002 _N. J. A. Sloane_
%E A000002 Minor edits to example and PARI code made by _M. F. Hasler_, May 07 2014
| 4ddd2d79d288754e934c5de5bbd844d6 |
A000003 | [
"M0196",
"N0073"
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] | Number of classes of primitive positive definite binary quadratic forms of discriminant D = -4n; or equivalently the class number of the quadratic order of discriminant D = -4n. | [
"A000003",
"A006643",
"A014599"
] | N. J. A. Sloane | [
"nonn",
"nice",
"easy"
] | 0 | 5 | oeisdata/seq/A000/A000003.seq | %I A000003 M0196 N0073 #82 Aug 14 2022 17:00:40
%S A000003 1,1,1,1,2,2,1,2,2,2,3,2,2,4,2,2,4,2,3,4,4,2,3,4,2,6,3,2,6,4,3,4,4,4,
%T A000003 6,4,2,6,4,4,8,4,3,6,4,4,5,4,4,6,6,4,6,6,4,8,4,2,9,4,6,8,4,4,8,8,3,8,
%U A000003 8,4,7,4,4,10,6,6,8,4,5,8,6,4,9,8,4,10,6,4,12,8,6,6,4,8,8,8,4,8,6,4
%N A000003 Number of classes of primitive positive definite binary quadratic forms of discriminant D = -4n; or equivalently the class number of the quadratic order of discriminant D = -4n.
%C A000003 From _Joerg Arndt_, Sep 02 2008: (Start)
%C A000003 It seems that 2*a(n) gives the degree of the minimal polynomial of (k_n)^2 where k_n is the n-th singular value, i.e., K(sqrt(1-k_n^2))/K(k_n)==sqrt(n) (and K is the elliptic integral of the first kind: K(x) := hypergeom([1/2,1/2],[1], x^2)).
%C A000003 Also, when setting K3(x)=hypergeom([1/3,2/3],[1], x^3) and solving for x such that K3((1-x^3)^(1/3))/K3(x)==sqrt(n), then the degree of the minimal polynomial of x^3 is every third term of this sequence, or so it seems. (End)
%C A000003 a(n) appears to be the degree of Klein's j-invariant j(sqrt(-n)) as an algebraic integer. - _Li Han_, Mar 02 2020
%D A000003 D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pages 20 and 231-234.[Dics means D = - Discriminant (see p. 223), and only squarefree cases appear on pp. 231-234, but not on p. 20. - _Wolfdieter Lang_, May 15 2021]
%D A000003 H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 514.
%D A000003 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000003 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000003 N. J. A. Sloane, <a href="/A000003/b000003.txt">Table of n, a(n) for n = 1..20000</a>
%H A000003 Harriet Fell, Morris Newman, Edward Ordman, <a href="https://dx.doi.org/10.6028/jres.067B.006">Tables of genera of groups of linear fractional transformations</a>, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
%H A000003 Daniel Shanks, <a href="https://doi.org/10.1090/S0025-5718-1960-0120203-6">On the Conjecture of Hardy & Littlewood concerning the Number of Primes of the Form n^2 + a</a>, Math. Comp. 14 (1960), 320-332. (Table 1 gives first 100 terms.)
%H A000003 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223295-5">Generalized Euler and class numbers</a>. Math. Comp. 21 (1967) 689-694.
%H A000003 D. Shanks, <a href="/A000003/a000003.pdf">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
%t A000003 a[1] = 1; a[n_] := (k0 = k /. FindRoot[EllipticK[1-k^2]/EllipticK[k^2] == Sqrt[n], {k, 1/2, 10^-10, 1}, WorkingPrecision -> 600, MaxIterations -> 100]; Exponent[ MinimalPolynomial[RootApproximant[k0^2, 24], x], x]/2); Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 1, 100}] (* _Jean-François Alcover_, Jul 21 2015, after _Joerg Arndt_ *)
%o A000003 (Magma) O1 := MaximalOrder(QuadraticField(D)); _,f := IsSquare(D div Discriminant(O1)); ClassNumber(sub<O1|f>);
%o A000003 (PARI) {a(n) = qfbclassno(-4*n)}; /* _Michael Somos_, Jul 16 1999 */
%Y A000003 See A014599 for discriminant -(4n-1).
%Y A000003 A006643 is a subsequence.
%K A000003 nonn,nice,easy
%O A000003 1,5
%A A000003 _N. J. A. Sloane_
| 463f83a694d856b16a52f15b756573eb |
A000004 | [
"M0000"
] | 128 | 2023-01-13T09:26:26 | [
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"0",
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"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0"
] | The zero sequence. | [
"A000004",
"A000007",
"A000012",
"A007395",
"A010701"
] | N. J. A. Sloane | [
"core",
"easy",
"nonn",
"mult"
] | 0 | 5 | oeisdata/seq/A000/A000004.seq | %I A000004 M0000 #128 Jan 13 2023 09:26:26
%S A000004 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A000004 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A000004 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A000004 The zero sequence.
%H A000004 N. J. A. Sloane, <a href="/A000004/b000004.txt">Table of n, a(n) for n = 0..1000</a> [Useful when <a href="/plot2.html">plotting one sequence against another</a>.]
%H A000004 Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H A000004 N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 10.
%H A000004 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H A000004 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000004 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%H A000004 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%F A000004 a(n) = 0 for all integer n.
%p A000004 A000004 := n->0;
%t A000004 a[ n_] := 0;
%t A000004 Table[0, {n, 100}] (* _Matthew House_, Jul 14 2015 *)
%t A000004 LinearRecurrence[{1},{0},102] (* _Ray Chandler_, Jul 15 2015 *)
%o A000004 (Magma) [ 0 : n in [0..100]];
%o A000004 (PARI) vector(100,n,0)
%o A000004 (R) rep(0,100)
%o A000004 (Haskell)
%o A000004 a000004 = const 0
%o A000004 a000004_list = repeat 0 -- _Reinhard Zumkeller_, May 07 2012
%o A000004 (Python) print([0 for n in range(102)]) # _Michael S. Branicky_, Apr 04 2022
%Y A000004 Cf. A000012 (all 1's), A007395 (all 2's), A010701 (all 3's).
%Y A000004 Cf. A000007(n) = 0^n: characteristic function of {0}.
%K A000004 core,easy,nonn,mult
%O A000004 0,1
%A A000004 _N. J. A. Sloane_
| 24945a4b17c49715e0c674e295fe2a63 |
A000005 | [
"M0246",
"N0086"
] | 564 | 2025-02-16T08:32:18 | [
"1",
"2",
"2",
"3",
"2",
"4",
"2",
"4",
"3",
"4",
"2",
"6",
"2",
"4",
"4",
"5",
"2",
"6",
"2",
"6",
"4",
"4",
"2",
"8",
"3",
"4",
"4",
"6",
"2",
"8",
"2",
"6",
"4",
"4",
"4",
"9",
"2",
"4",
"4",
"8",
"2",
"8",
"2",
"6",
"6",
"4",
"2",
"10",
"3",
"6",
"4",
"6",
"2",
"8",
"4",
"8",
"4",
"4",
"2",
"12",
"2",
"4",
"6",
"7",
"4",
"8",
"2",
"6",
"4",
"8",
"2",
"12",
"2",
"4",
"6",
"6",
"4",
"8",
"2",
"10",
"5",
"4",
"2",
"12",
"4",
"4",
"4",
"8",
"2",
"12",
"4",
"6",
"4",
"4",
"4",
"12",
"2",
"6",
"6",
"9",
"2",
"8",
"2",
"8"
] | d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n. | [
"A000005",
"A000010",
"A000203",
"A001227",
"A001826",
"A001842",
"A002182",
"A002183",
"A005237",
"A005238",
"A006218",
"A006558",
"A006601",
"A007425",
"A007426",
"A007427",
"A019273",
"A027750",
"A034296",
"A034836",
"A038548",
"A039665",
"A049051",
"A049820",
"A051731",
"A061017",
"A066446",
"A091202",
"A091220",
"A098198",
"A106737",
"A115361",
"A127093",
"A129372",
"A129510",
"A143319",
"A156552",
"A159933",
"A159934",
"A163280",
"A183030",
"A183031",
"A183063",
"A237665",
"A263730"
] | N. J. A. Sloane | [
"easy",
"core",
"nonn",
"nice",
"mult",
"hear"
] | 0 | 5 | oeisdata/seq/A000/A000005.seq | %I A000005 M0246 N0086 #564 Feb 16 2025 08:32:18
%S A000005 1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3,4,4,6,2,8,2,6,4,4,
%T A000005 4,9,2,4,4,8,2,8,2,6,6,4,2,10,3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,
%U A000005 6,4,8,2,12,2,4,6,6,4,8,2,10,5,4,2,12,4,4,4,8,2,12,4,6,4,4,4,12,2,6,6,9,2,8,2,8
%N A000005 d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.
%C A000005 If the canonical factorization of n into prime powers is Product p^e(p) then d(n) = Product (e(p) + 1). More generally, for k > 0, sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1) is the sum of the k-th powers of the divisors of n.
%C A000005 Number of ways to write n as n = x*y, 1 <= x <= n, 1 <= y <= n. For number of unordered solutions to x*y=n, see A038548.
%C A000005 Note that d(n) is not the number of Pythagorean triangles with radius of the inscribed circle equal to n (that is A078644). For number of primitive Pythagorean triangles having inradius n, see A068068(n).
%C A000005 Number of factors in the factorization of the polynomial x^n-1 over the integers. - _T. D. Noe_, Apr 16 2003
%C A000005 Also equal to the number of partitions p of n such that all the parts have the same cardinality, i.e., max(p)=min(p). - _Giovanni Resta_, Feb 06 2006
%C A000005 Equals A127093 as an infinite lower triangular matrix * the harmonic series, [1/1, 1/2, 1/3, ...]. - _Gary W. Adamson_, May 10 2007
%C A000005 For odd n, this is the number of partitions of n into consecutive integers. Proof: For n = 1, clearly true. For n = 2k + 1, k >= 1, map each (necessarily odd) divisor to such a partition as follows: For 1 and n, map k + (k+1) and n, respectively. For any remaining divisor d <= sqrt(n), map (n/d - (d-1)/2) + ... + (n/d - 1) + (n/d) + (n/d + 1) + ... + (n/d + (d-1)/2) {i.e., n/d plus (d-1)/2 pairs each summing to 2n/d}. For any remaining divisor d > sqrt(n), map ((d-1)/2 - (n/d - 1)) + ... + ((d-1)/2 - 1) + (d-1)/2 + (d+1)/2 + ((d+1)/2 + 1) + ... + ((d+1)/2 + (n/d - 1)) {i.e., n/d pairs each summing to d}. As all such partitions must be of one of the above forms, the 1-to-1 correspondence and proof is complete. - _Rick L. Shepherd_, Apr 20 2008
%C A000005 Number of subgroups of the cyclic group of order n. - _Benoit Jubin_, Apr 29 2008
%C A000005 Equals row sums of triangle A143319. - _Gary W. Adamson_, Aug 07 2008
%C A000005 Equals row sums of triangle A159934, equivalent to generating a(n) by convolving A000005 prefaced with a 1; (1, 1, 2, 2, 3, 2, ...) with the INVERTi transform of A000005, (A159933): (1, 1,-1, 0, -1, 2, ...). Example: a(6) = 4 = (1, 1, 2, 2, 3, 2) dot (2, -1, 0, -1, 1, 1) = (2, -1, 0, -2, 3, 2) = 4. - _Gary W. Adamson_, Apr 26 2009
%C A000005 Number of times n appears in an n X n multiplication table. - _Dominick Cancilla_, Aug 02 2010
%C A000005 Number of k >= 0 such that (k^2 + k*n + k)/(k + 1) is an integer. - _Juri-Stepan Gerasimov_, Oct 25 2015
%C A000005 The only numbers k such that tau(k) >= k/2 are 1,2,3,4,6,8,12. - _Michael De Vlieger_, Dec 14 2016
%C A000005 a(n) is also the number of partitions of 2*n into equal parts, minus the number of partitions of 2*n into consecutive parts. - _Omar E. Pol_, May 03 2017
%C A000005 From _Tomohiro Yamada_, Oct 27 2020: (Start)
%C A000005 Let k(n) = log d(n)*log log n/(log 2 * log n), then lim sup k(n) = 1 (Hardy and Wright, Chapter 18, Theorem 317) and k(n) <= k(6983776800) = 1.537939... (the constant A280235) for every n (Nicolas and Robin, 1983).
%C A000005 There exist infinitely many n such that d(n) = d(n+1) (Heath-Brown, 1984). The number of such integers n <= x is at least c*x/(log log x)^3 (Hildebrand, 1987) but at most O(x/sqrt(log log x)) (Erdős, Carl Pomerance and Sárközy, 1987).
%C A000005 (End)
%C A000005 Number of 2D grids of n congruent rectangles with two different side lengths, in a rectangle, modulo rotation (cf. A038548 for squares instead of rectangles). Also number of ways to arrange n identical objects in a rectangle (NOT modulo rotation, cf. A038548 for modulo rotation); cf. A007425 and A140773 for the 3D case. - _Manfred Boergens_, Jun 08 2021
%D A000005 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A000005 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
%D A000005 G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 345, Exercise XXI(16). MR0121327 (22 #12066)
%D A000005 G. H. Hardy and E. M. Wright, revised by D. R. Heath-Brown and J. H. Silverman, An Introduction to the Theory of Numbers, 6th ed., Oxford Univ. Press, 2008.
%D A000005 K. Knopp, Theory and Application of Infinite Series, Blackie, London, 1951, p. 451.
%D A000005 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Chap. II. (For inequalities, etc.)
%D A000005 S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. Has many references to this sequence. - _N. J. A. Sloane_, Jun 02 2014
%D A000005 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000005 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000005 B. Spearman and K. S. Williams, Handbook of Estimates in the Theory of Numbers, Carleton Math. Lecture Note Series No. 14, 1975; see p. 2.1.
%D A000005 E. C. Titchmarsh, The Theory of Functions, Oxford, 1938, p. 160.
%D A000005 Terence Tao, Poincaré's Legacies, Part I, Amer. Math. Soc., 2009, see pp. 31ff for upper bounds on d(n).
%H A000005 Daniel Forgues, <a href="/A000005/b000005.txt">Table of n, a(n) for n = 1..100000</a> (first 10000 terms from N. J. A. Sloane)
%H A000005 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy, requires Flash plugin].
%H A000005 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/s/s42andrews.html">Some debts I owe</a>, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.1).
%H A000005 J. Bell, <a href="https://jordanbell.info/writing/2023/02/22/lambert-series-analytic-number-theory.html">Lambert series in analytic number theory</a>
%H A000005 R. Bellman and H. N. Shapiro, <a href="http://www.jstor.org/stable/1969281">On a problem in additive number theory</a>, Annals Math., 49 (1948), 333-340. [From _N. J. A. Sloane_, Mar 12 2009]
%H A000005 Henry Bottomley, <a href="/A000005/a000005.gif">Illustration of initial terms</a>
%H A000005 D. M. Bressoud and M. V. Subbarao, <a href="http://dx.doi.org/10.4153/CMB-1984-022-5">On Uchimura's connection between partitions and the number of divisors</a>, Can. Math. Bull. 27, 143-145 (1984). Zbl 0536.10013.
%H A000005 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=Tau">Number of divisors</a>
%H A000005 Imanuel Chen and Michael Z. Spivey, <a href="http://soundideas.pugetsound.edu/summer_research/238">Integral Generalized Binomial Coefficients of Multiplicative Functions</a>, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
%H A000005 Jimmy Devillet and Gergely Kiss, <a href="https://arxiv.org/abs/1806.02073">Characterizations of biselective operations</a>, arXiv:1806.02073 [math.RA], 2018.
%H A000005 P. Erdős and L. Mirsky, <a href="http://www.renyi.hu/~p_erdos/1952-12.pdf">The distribution of values of the divisor function d(n)</a>, Proc. London Math. Soc. 2 (1952), pp. 257-271.
%H A000005 Paul Erdős, Carl Pomerance and András Sárközy, <a href="https://doi.org/10.1090/S0002-9939-1987-0897061-6">On locally repeated values of certain arithmetic functions, III</a>, Proc. Amer. Math. Soc. 101 (1987), 1-7.
%H A000005 C. R. Fletcher, <a href="http://www.jstor.org/stable/3615885">Rings of small order</a>, Math. Gaz. vol. 64, p. 13, 1980.
%H A000005 Robbert Fokkink and Jan van Neerven, <a href="https://www.nieuwarchief.nl/serie5/pdf/naw5-2003-04-3-269.pdf">Problemen/UWC</a> (in Dutch)
%H A000005 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors ...</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000005 D. R. Heath-Brown, <a href="https://doi.org/10.1112/S0025579300010743">The divisor function at consecutive integers</a>, Mathematika 31 (1984), 141-149.
%H A000005 Adolf Hildebrand, <a href="https://projecteuclid.org/euclid.pjm/1102690578">The divisor function at consecutive integers</a>, Pacific J. Math. 129 (1987), 307-319.
%H A000005 J. J. Holt and J. W. Jones, <a href="https://web.archive.org/web/20190310134052/http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/divis4.html">Counting Divisors</a>, Discovering Number Theory, Section 1.4.
%H A000005 P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1919), 75-113.
%H A000005 M. Maia and M. Mendez, <a href="https://arxiv.org/abs/math/0503436">On the arithmetic product of combinatorial species</a>, arXiv:math/0503436 [math.CO], 2005.
%H A000005 R. G. Martinez, Jr., The Factor Zone, <a href="http://factorzone.tripod.com/factors.htm">Number of Factors for 1 through 600</a>.
%H A000005 Math Forum, <a href="http://mathforum.org/library/drmath/view/55741.html">Divisor Counting</a>.
%H A000005 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3213216/a-question-on-discrete-fourier-transform-of-some-function">A question on discrete Fourier Transform of some function</a>
%H A000005 K. Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n)</a>.
%H A000005 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2014.10.009">A new look on the generating function for the number of divisors</a>, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
%H A000005 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, corollary 2.1.
%H A000005 Matthew Parker, <a href="https://oeis.org/A000005/a000005_25M.7z">The first 25 million terms (7-Zip compressed file)</a>.
%H A000005 Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/MAA/07-Sequence%20Pictures/mathgames_12_08_03.html">Sequence Pictures</a>, Math Games column, Dec 08 2003.
%H A000005 Ed Pegg, Jr., <a href="/A000043/a000043_2.pdf">Sequence Pictures</a>, Math Games column, Dec 08 2003. [Cached copy, with permission (pdf only)]
%H A000005 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv01.jpg">Illustration of initial terms: figure 1</a>, <a href="http://www.polprimos.com/imagenespub/poldiv02.jpg">figure 2</a>, <a href="http://www.polprimos.com/imagenespub/poldiv03.jpg">figure 3</a>, <a href="http://www.polprimos.com/imagenespub/poldiv04.jpg">figure 4</a>, <a href="http://www.polprimos.com/imagenespub/poldiv3v.jpg">figure 5</a>, (2009), <a href="http://www.polprimos.com/imagenespub/poldiv13.jpg">figure 6 (a, b, c)</a>, (2013)
%H A000005 S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper8/page1.htm">On The Number Of Divisors Of A Number</a>.
%H A000005 H. B. Reiter, <a href="https://webpages.charlotte.edu/~hbreiter/m6105/Divisors.pdf">Counting Divisors</a>.
%H A000005 W. Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>.
%H A000005 Terence Tao, <a href="https://terrytao.wordpress.com/wp-content/uploads/2009/01/whatsnew.pdf">Poincaré's Legacies: pages from year two of a mathematical blog</a>, see page 59.
%H A000005 E. C. Titchmarsh, <a href="https://doi.org/10.1112/jlms/s1-13.4.248">On a series of Lambert type</a>, J. London Math. Soc., 13 (1938), 248-253.
%H A000005 Keisuke Uchimura, <a href="http://dx.doi.org/10.1016/0097-3165(81)90009-1">An identity for the divisor generating function arising from sorting theory</a>, J. Combin. Theory Ser. A 31 (1981), no. 2, 131--135. MR0629588 (82k:05015)
%H A000005 Wang Zheng Bing, Robert Fokkink and Wan Fokkink, <a href="http://www.jstor.org/stable/2974956">A Relation Between Partitions and the Number of Divisors</a>, Am. Math. Monthly, 102 (Apr., 1995), no. 4, 345-347.
%H A000005 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialNumber.html">Binomial Number</a>, <a href="https://mathworld.wolfram.com/DirichletSeriesGeneratingFunction.html">Dirichlet Series Generating Function</a>, <a href="https://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>, and <a href="https://mathworld.wolfram.com/MoebiusTransform.html">Moebius Transform</a>.
%H A000005 Wikipedia, <a href="http://www.wikipedia.org/wiki/Table_of_divisors">Table of divisors</a>.
%H A000005 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/Divisors/03/02">Divisors of first 50 numbers</a>
%H A000005 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000005 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A000005 If n is written as 2^z*3^y*5^x*7^w*11^v*... then a(n)=(z+1)*(y+1)*(x+1)*(w+1)*(v+1)*...
%F A000005 a(n) = 2 iff n is prime.
%F A000005 G.f.: Sum_{n >= 1} a(n) x^n = Sum_{k>0} x^k/(1-x^k). This is usually called THE Lambert series (see Knopp, Titchmarsh).
%F A000005 a(n) = A083888(n) + A083889(n) + A083890(n) + A083891(n) + A083892(n) + A083893(n) + A083894(n) + A083895(n) + A083896(n).
%F A000005 a(n) = A083910(n) + A083911(n) + A083912(n) + A083913(n) + A083914(n) + A083915(n) + A083916(n) + A083917(n) + A083918(n) + A083919(n).
%F A000005 Multiplicative with a(p^e) = e+1. - _David W. Wilson_, Aug 01 2001
%F A000005 a(n) <= 2 sqrt(n) [see Mitrinovich, p. 39, also A046522].
%F A000005 a(n) is odd iff n is a square. - _Reinhard Zumkeller_, Dec 29 2001
%F A000005 a(n) = Sum_{k=1..n} f(k, n) where f(k, n) = 1 if k divides n, 0 otherwise (Mobius transform of A000012). Equivalently, f(k, n) = (1/k)*Sum_{l=1..k} z(k, l)^n with z(k, l) the k-th roots of unity. - _Ralf Stephan_, Dec 25 2002
%F A000005 G.f.: Sum_{k>0} ((-1)^(k+1) * x^(k * (k + 1)/2) / ((1 - x^k) * Product_{i=1..k} (1 - x^i))). - _Michael Somos_, Apr 27 2003
%F A000005 a(n) = n - Sum_{k=1..n} (ceiling(n/k) - floor(n/k)). - _Benoit Cloitre_, May 11 2003
%F A000005 a(n) = A032741(n) + 1 = A062011(n)/2 = A054519(n) - A054519(n-1) = A006218(n) - A006218(n-1) = 1 + Sum_{k=1..n-1} A051950(k+1). - _Ralf Stephan_, Mar 26 2004
%F A000005 G.f.: Sum_{k>0} x^(k^2)*(1+x^k)/(1-x^k). Dirichlet g.f.: zeta(s)^2. - _Michael Somos_, Apr 05 2003
%F A000005 Sequence = M*V where M = A129372 as an infinite lower triangular matrix and V = ruler sequence A001511 as a vector: [1, 2, 1, 3, 1, 2, 1, 4, ...]. - _Gary W. Adamson_, Apr 15 2007
%F A000005 Sequence = M*V, where M = A115361 is an infinite lower triangular matrix and V = A001227, the number of odd divisors of n, is a vector: [1, 1, 2, 1, 2, 2, 2, ...]. - _Gary W. Adamson_, Apr 15 2007
%F A000005 Row sums of triangle A051731. - _Gary W. Adamson_, Nov 02 2007
%F A000005 Sum_{n>0} a(n)/(n^n) = Sum_{n>0, m>0} 1/(n*m). - _Gerald McGarvey_, Dec 15 2007
%F A000005 Logarithmic g.f.: Sum_{n>=1} a(n)/n * x^n = -log( Product_{n>=1} (1-x^n)^(1/n) ). - _Joerg Arndt_, May 03 2008
%F A000005 a(n) = Sum_{k=1..n} (floor(n/k) - floor((n-1)/k)). - _Enrique Pérez Herrero_, Aug 27 2009
%F A000005 a(s) = 2^omega(s), if s > 1 is a squarefree number (A005117) and omega(s) is: A001221. - _Enrique Pérez Herrero_, Sep 08 2009
%F A000005 a(n) = A048691(n) - A055205(n). - _Reinhard Zumkeller_, Dec 08 2009
%F A000005 For n > 1, a(n) = 2 + Sum_{k=2..n-1} floor((cos(Pi*n/k))^2). And floor((cos(Pi*n/k))^2) = floor(1/4 * e^(-(2*i*Pi*n)/k) + 1/4 * e^((2*i*Pi*n)/k) + 1/2). - _Eric Desbiaux_, Mar 09 2010, corrected Apr 16 2011
%F A000005 a(n) = 1 + Sum_{k=1..n} (floor(2^n/(2^k-1)) mod 2) for every n. - Fabio Civolani (civox(AT)tiscali.it), Mar 12 2010
%F A000005 From _Vladimir Shevelev_, May 22 2010: (Start)
%F A000005 (Sum_{d|n} a(d))^2 = Sum_{d|n} a(d)^3 (J. Liouville).
%F A000005 Sum_{d|n} A008836(d)*a(d)^2 = A008836(n)*Sum_{d|n} a(d). (End)
%F A000005 a(n) = sigma_0(n) = 1 + Sum_{m>=2} Sum_{r>=1} (1/m^(r+1))*Sum_{j=1..m-1} Sum_{k=0..m^(r+1)-1} e^(2*k*Pi*i*(n+(m-j)*m^r)/m^(r+1)). - _A. Neves_, Oct 04 2010
%F A000005 a(n) = 2*A038548(n) - A010052(n). - _Reinhard Zumkeller_, Mar 08 2013
%F A000005 Sum_{n>=1} a(n)*q^n = (log(1-q) + psi_q(1)) / log(q), where psi_q(z) is the q-digamma function. - _Vladimir Reshetnikov_, Apr 23 2013
%F A000005 a(n) = Product_{k = 1..A001221(n)} (A124010(n,k) + 1). - _Reinhard Zumkeller_, Jul 12 2013
%F A000005 a(n) = Sum_{k=1..n} A238133(k)*A000041(n-k). - _Mircea Merca_, Feb 18 2013
%F A000005 G.f.: Sum_{k>=1} Sum_{j>=1} x^(j*k). - _Mats Granvik_, Jun 15 2013
%F A000005 The formula above is obtained by expanding the Lambert series Sum_{k>=1} x^k/(1-x^k). - _Joerg Arndt_, Mar 12 2014
%F A000005 G.f.: Sum_{n>=1} Sum_{d|n} ( -log(1 - x^(n/d)) )^d / d!. - _Paul D. Hanna_, Aug 21 2014
%F A000005 2*Pi*a(n) = Sum_{m=1..n} Integral_{x=0..2*Pi} r^(m-n)( cos((m-n)*x)-r^m cos(n*x) )/( 1+r^(2*m)-2r^m cos(m*x) )dx, 0 < r < 1 a free parameter. This formula is obtained as the sum of the residues of the Lambert series Sum_{k>=1} x^k/(1-x^k). - _Seiichi Kirikami_, Oct 22 2015
%F A000005 a(n) = A091220(A091202(n)) = A106737(A156552(n)). - _Antti Karttunen_, circa 2004 & Mar 06 2017
%F A000005 a(n) = A034296(n) - A237665(n+1) [Wang, Fokkink, Fokkink]. - _George Beck_, May 06 2017
%F A000005 G.f.: 2*x/(1-x) - Sum_{k>0} x^k*(1-2*x^k)/(1-x^k). - _Mamuka Jibladze_, Aug 29 2018
%F A000005 a(n) = Sum_{k=1..n} 1/phi(n / gcd(n, k)). - _Daniel Suteu_, Nov 05 2018
%F A000005 a(k*n) = a(n)*(f(k,n)+2)/(f(k,n)+1), where f(k,n) is the exponent of the highest power of k dividing n and k is prime. - _Gary Detlefs_, Feb 08 2019
%F A000005 a(n) = 2*log(p(n))/log(n), n > 1, where p(n)= the product of the factors of n = A007955(n). - _Gary Detlefs_, Feb 15 2019
%F A000005 a(n) = (1/n) * Sum_{k=1..n} sigma(gcd(n,k)), where sigma(n) = sum of divisors of n. - _Orges Leka_, May 09 2019
%F A000005 a(n) = A001227(n)*(A007814(n) + 1) = A001227(n)*A001511(n). - _Ivan N. Ianakiev_, Nov 14 2019
%F A000005 From _Richard L. Ollerton_, May 11 2021: (Start)
%F A000005 a(n) = A038040(n) / n = (1/n)*Sum_{d|n} phi(d)*sigma(n/d), where phi = A000010 and sigma = A000203.
%F A000005 a(n) = (1/n)*Sum_{k=1..n} phi(gcd(n,k))*sigma(n/gcd(n,k))/phi(n/gcd(n,k)). (End)
%F A000005 From _Ridouane Oudra_, Nov 12 2021: (Start)
%F A000005 a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*cos(2*k*n*Pi/j);
%F A000005 a(n) = Sum_{j=1..n} Sum_{k=1..j} (1/j)*e^(2*k*n*Pi*i/j), where i^2=-1. (End)
%e A000005 G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 2*x^7 + 4*x^8 + 3*x^9 + ...
%p A000005 with(numtheory): A000005 := tau; [ seq(tau(n), n=1..100) ];
%t A000005 Table[DivisorSigma[0, n], {n, 100}] (* _Enrique Pérez Herrero_, Aug 27 2009 *)
%t A000005 CoefficientList[Series[(Log[1 - q] + QPolyGamma[1, q])/(q Log[q]), {q, 0, 100}], q] (* _Vladimir Reshetnikov_, Apr 23 2013 *)
%t A000005 a[ n_] := SeriesCoefficient[ (QPolyGamma[ 1, q] + Log[1 - q]) / Log[q], {q, 0, Abs@n}]; (* _Michael Somos_, Apr 25 2013 *)
%t A000005 a[ n_] := SeriesCoefficient[ q/(1 - q)^2 QHypergeometricPFQ[ {q, q}, {q^2, q^2}, q, q^2], {q, 0, Abs@n}]; (* _Michael Somos_, Mar 05 2014 *)
%t A000005 a[n_] := SeriesCoefficient[q/(1 - q) QHypergeometricPFQ[{q, q}, {q^2}, q, q], {q, 0, Abs@n}] (* _Mats Granvik_, Apr 15 2015 *)
%t A000005 With[{M=500},CoefficientList[Series[(2x)/(1-x)-Sum[x^k (1-2x^k)/(1-x^k),{k,M}],{x,0,M}],x]] (* _Mamuka Jibladze_, Aug 31 2018 *)
%o A000005 (PARI) {a(n) = if( n==0, 0, numdiv(n))}; /* _Michael Somos_, Apr 27 2003 */
%o A000005 (PARI) {a(n) = n=abs(n); if( n<1, 0, direuler( p=2, n, 1 / (1 - X)^2)[n])}; /* _Michael Somos_, Apr 27 2003 */
%o A000005 (PARI) {a(n)=polcoeff(sum(m=1, n+1, sumdiv(m, d, (-log(1-x^(m/d) +x*O(x^n) ))^d/d!)), n)} \\ _Paul D. Hanna_, Aug 21 2014
%o A000005 (Magma) [ NumberOfDivisors(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
%o A000005 (MuPAD) numlib::tau (n)$ n=1..90 // _Zerinvary Lajos_, May 13 2008
%o A000005 (Sage) [sigma(n, 0) for n in range(1, 105)] # _Zerinvary Lajos_, Jun 04 2009
%o A000005 (Haskell)
%o A000005 divisors 1 = [1]
%o A000005 divisors n = (1:filter ((==0) . rem n)
%o A000005 [2..n `div` 2]) ++ [n]
%o A000005 a = length . divisors
%o A000005 -- _James Spahlinger_, Oct 07 2012
%o A000005 (Haskell)
%o A000005 a000005 = product . map (+ 1) . a124010_row -- _Reinhard Zumkeller_, Jul 12 2013
%o A000005 (Python)
%o A000005 from sympy import divisor_count
%o A000005 for n in range(1, 20): print(divisor_count(n), end=', ') # _Stefano Spezia_, Nov 05 2018
%o A000005 (GAP) List([1..150],n->Tau(n)); # _Muniru A Asiru_, Mar 05 2019
%o A000005 (Julia)
%o A000005 function tau(n)
%o A000005 i = 2; num = 1
%o A000005 while i * i <= n
%o A000005 if rem(n, i) == 0
%o A000005 e = 0
%o A000005 while rem(n, i) == 0
%o A000005 e += 1
%o A000005 n = div(n, i)
%o A000005 end
%o A000005 num *= e + 1
%o A000005 end
%o A000005 i += 1
%o A000005 end
%o A000005 return n > 1 ? num + num : num
%o A000005 end
%o A000005 println([tau(n) for n in 1:104]) # _Peter Luschny_, Sep 03 2023
%Y A000005 See A002183, A002182 for records. See A000203 for the sum-of-divisors function sigma(n).
%Y A000005 For partial sums see A006218.
%Y A000005 Cf. A007427 (Dirichlet Inverse), A001227, A005237, A005238, A006601, A006558, A019273, A039665, A049051, A001826, A001842, A049820, A051731, A066446, A106737, A129510, A115361, A129372, A127093, A143319, A061017, A091202, A091220, A156552, A159933, A159934, A027750, A163280, A183063, A263730, A034296, A237665.
%Y A000005 Factorizations into given number of factors: writing n = x*y (A038548, unordered, A000005, ordered), n = x*y*z (A034836, unordered, A007425, ordered), n = w*x*y*z (A007426, ordered).
%Y A000005 Cf. A000010.
%Y A000005 Cf. A098198 (Dgf at s=2), A183030 (Dgf at s=3), A183031 (Dgf at s=3).
%K A000005 easy,core,nonn,nice,mult,hear
%O A000005 1,2
%A A000005 _N. J. A. Sloane_
%E A000005 Incorrect formula deleted by _Ridouane Oudra_, Oct 28 2021
| 24610e6dc474e31b8d5a4c1c98791d9c |
A000006 | [
"M0259",
"N0092"
] | 51 | 2020-06-25T15:04:53 | [
"1",
"1",
"2",
"2",
"3",
"3",
"4",
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"5",
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"17",
"17",
"17",
"17",
"17",
"18",
"18",
"18",
"18",
"18"
] | Integer part of square root of n-th prime. | [
"A000006",
"A000196",
"A014085",
"A048766",
"A263846"
] | N. J. A. Sloane | [
"nonn",
"easy",
"nice"
] | 0 | 5 | oeisdata/seq/A000/A000006.seq | %I A000006 M0259 N0092 #51 Jun 25 2020 15:04:53
%S A000006 1,1,2,2,3,3,4,4,4,5,5,6,6,6,6,7,7,7,8,8,8,8,9,9,9,10,10,10,10,10,11,
%T A000006 11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,14,14,15,15,15,15,15,15,
%U A000006 16,16,16,16,16,16,16,17,17,17,17,17,18,18,18,18,18
%N A000006 Integer part of square root of n-th prime.
%C A000006 Conjecture: No two successive terms in the sequence differ by more than 1. Proof of this would prove the converse of the theorem that every prime is surrounded by two consecutive squares, namely |sqrt(p)|^2 and (|sqrt(p)|+1)^2. - _Cino Hilliard_, Jan 22 2003
%C A000006 Equals the number of squares less than prime(n). Cf. A014689. - _Zak Seidov_ Nov 04 2007
%C A000006 The above conjecture is Legendre's conjecture that for n > 0 there is always a prime between n^2 and (n+1)^2. See A014085, number of primes between two consecutive squares, which is also the number of times n is repeated in the present sequence. - _Jean-Christophe Hervé_, Oct 25 2013
%D A000006 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
%D A000006 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000006 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000006 T. D. Noe, <a href="/A000006/b000006.txt">Table of n, a(n) for n = 1..10000</a>
%H A000006 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A000006 Matthew Parker, <a href="https://oeis.org/A000006/a000006_100M.7z">The first 100 million terms (7-Zip compressed file)</a>
%F A000006 a(n) = A000196(A000040(n)). - _Reinhard Zumkeller_, Mar 24 2012
%t A000006 a[n_] := IntegerPart[Sqrt[Prime[n]]]
%t A000006 IntegerPart[Sqrt[#]]&/@Prime[Range[80]] (* _Harvey P. Dale_, Mar 06 2012 *)
%o A000006 (PARI) (a(n)=sqrtint(prime(n))); vector(100,n,a(n)) \\ Edited by _M. F. Hasler_, Oct 19 2018
%o A000006 (PARI) apply(sqrtint,primes(100)) \\ _Charles R Greathouse IV_, Apr 26 2012
%o A000006 (PARI) apply( A000006=n->sqrtint(prime(n)), [1..100]) \\ _M. F. Hasler_, Oct 19 2018
%o A000006 (Haskell) a000006 = a000196 . a000040 -- _Reinhard Zumkeller_, Mar 24 2012
%o A000006 (Python)
%o A000006 from sympy import sieve
%o A000006 A000006 = lambda n: int(sieve[n]**.5)
%o A000006 print([A000006(n) for n in range(1,100+1)])
%o A000006 # _Albert Lahat_, Jun 25 2020
%Y A000006 Cf. A014085.
%Y A000006 See also A263846 (floor of cube root of prime(n)), A000196 (floor of sqrt(n)), A048766 (floor of cube root of n).
%K A000006 nonn,easy,nice
%O A000006 1,3
%A A000006 _N. J. A. Sloane_
| 2d9667bc102d2236961746cffb1a1383 |
A000007 | [
"M0002"
] | 296 | 2025-02-16T08:32:18 | [
"1",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
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"0",
"0",
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"0",
"0",
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"0",
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"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0",
"0"
] | The characteristic function of {0}: a(n) = 0^n. | [
"A000007",
"A000012",
"A027641",
"A057427",
"A059841",
"A063524",
"A074909",
"A079978",
"A079979",
"A079998",
"A082784",
"A121262",
"A185012",
"A185013",
"A185014",
"A185015",
"A185016",
"A185017"
] | N. J. A. Sloane | [
"core",
"nonn",
"mult",
"cons",
"easy"
] | 0 | 5 | oeisdata/seq/A000/A000007.seq | %I A000007 M0002 #296 Feb 16 2025 08:32:18
%S A000007 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A000007 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A000007 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A000007 The characteristic function of {0}: a(n) = 0^n.
%C A000007 Changing the offset to 1 gives the arithmetical function a(1) = 1, a(n) = 0 for n > 1, the identity function for Dirichlet multiplication (see Apostol). - _N. J. A. Sloane_
%C A000007 Changing the offset to 1 makes this the decimal expansion of 1. - _N. J. A. Sloane_, Nov 13 2014
%C A000007 Hankel transform (see A001906 for definition) of A000007 (powers of 0), A000012 (powers of 1), A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001018 (powers of 8), A001019 (powers of 9), A011557 (powers of 10), A001020 (powers of 11), etc. - _Philippe Deléham_, Jul 07 2005
%C A000007 This is the identity sequence with respect to convolution. - _David W. Wilson_, Oct 30 2006
%C A000007 a(A000004(n)) = 1; a(A000027(n)) = 0. - _Reinhard Zumkeller_, Oct 12 2008
%C A000007 The alternating sum of the n-th row of Pascal's triangle gives the characteristic function of 0, a(n) = 0^n. - _Daniel Forgues_, May 25 2010
%C A000007 The number of maximal self-avoiding walks from the NW to SW corners of a 1 X n grid. - _Sean A. Irvine_, Nov 19 2010
%C A000007 Historically there has been some disagreement as to whether 0^0 = 1. Graphing x^0 seems to support that conclusion, but graphing 0^x instead suggests that 0^0 = 0. Euler and Knuth have argued in favor of 0^0 = 1. For some calculators, 0^0 triggers an error, while in Mathematica, 0^0 is Indeterminate. - _Alonso del Arte_, Nov 15 2011
%C A000007 Another consequence of changing the offset to 1 is that then this sequence can be described as the sum of Moebius mu(d) for the divisors d of n. - _Alonso del Arte_, Nov 28 2011
%C A000007 With the convention 0^0 = 1, 0^n = 0 for n > 0, the sequence a(n) = 0^|n-k|, which equals 1 when n = k and is 0 for n >= 0, has g.f. x^k. A000007 is the case k = 0. - _George F. Johnson_, Mar 08 2013
%C A000007 A fixed point of the run length transform. - _Chai Wah Wu_, Oct 21 2016
%D A000007 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.
%D A000007 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000007 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A000007 David Wasserman, <a href="/A000007/b000007.txt">Table of n, a(n) for n = 0..1000</a>
%H A000007 Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A000007 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry2/barry231.html">A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.4.
%H A000007 Dr. Math, <a href="http://mathforum.org/dr.math/faq/faq.0.to.0.power.html">0^0 (zero to the zero power)</a>
%H A000007 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000007 Donald E. Knuth, <a href="http://arxiv.org/abs/math/9205211">Two notes on notation</a>, arXiv:math/9205211 [math.HO], 1992. See page 6 on 0^0.
%H A000007 Robert Price, <a href="/A000007/a000007.txt">Comments on A000007</a>, Jan 27 2016
%H A000007 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A000007 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A000007 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000007 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H A000007 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A000007 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A000007 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A000007 Multiplicative with a(p^e) = 0. - _David W. Wilson_, Sep 01 2001
%F A000007 a(n) = floor(1/(n + 1)). - _Franz Vrabec_, Aug 24 2005
%F A000007 As a function of Bernoulli numbers (cf. A027641: (1, -1/2, 1/6, 0, -1/30, ...)), triangle A074909 (the beheaded Pascal's triangle) * B_n as a vector = [1, 0, 0, 0, 0, ...]. - _Gary W. Adamson_, Mar 05 2012
%F A000007 a(n) = Sum_{k = 0..n} exp(2*Pi*i*k/(n+1)) is the sum of the (n+1)th roots of unity. - _Franz Vrabec_, Nov 09 2012
%F A000007 a(n) = (1-(-1)^(2^n))/2. - _Luce ETIENNE_, May 05 2015
%F A000007 a(n) = 1 - A057427(n). - _Alois P. Heinz_, Jan 20 2016
%F A000007 From _Ilya Gutkovskiy_, Sep 02 2016: (Start)
%F A000007 Binomial transform of A033999.
%F A000007 Inverse binomial transform of A000012. (End)
%p A000007 A000007 := proc(n) if n = 0 then 1 else 0 fi end: seq(A000007(n), n=0..20);
%p A000007 spec := [A, {A=Z} ]: seq(combstruct[count](spec, size=n+1), n=0..20);
%t A000007 Table[If[n == 0, 1, 0], {n, 0, 99}]
%t A000007 Table[Boole[n == 0], {n, 0, 99}] (* _Michael Somos_, Aug 25 2012 *)
%t A000007 Join[{1},LinearRecurrence[{1},{0},102]] (* _Ray Chandler_, Jul 30 2015 *)
%t A000007 PadRight[{1},120,0] (* _Harvey P. Dale_, Jul 18 2024 *)
%o A000007 (PARI) {a(n) = !n};
%o A000007 (Magma) [1] cat [0:n in [1..100]]; // Sergei Haller, Dec 21 2006
%o A000007 (Haskell)
%o A000007 a000007 = (0 ^)
%o A000007 a000007_list = 1 : repeat 0
%o A000007 -- _Reinhard Zumkeller_, May 07 2012, Mar 27 2012
%o A000007 (Python)
%o A000007 def A000007(n): return int(n==0) # _Chai Wah Wu_, Feb 04 2022
%Y A000007 Characteristic function of {g}: this sequence (g = 0), A063524 (g = 1), A185012 (g = 2), A185013 (g = 3), A185014 (g = 4), A185015 (g = 5), A185016 (g = 6), A185017 (g = 7). - _Jason Kimberley_, Oct 14 2011
%Y A000007 Characteristic function of multiples of g: this sequence (g = 0), A000012 (g = 1), A059841 (g = 2), A079978 (g = 3), A121262 (g = 4), A079998 (g = 5), A079979 (g = 6), A082784 (g = 7). - _Jason Kimberley_, Oct 14 2011
%Y A000007 Cf. A074909, A027641, A057427.
%K A000007 core,nonn,mult,cons,easy
%O A000007 0,1
%A A000007 _N. J. A. Sloane_
| 021497da56c19e024fd6bc7430368377 |
A000008 | [
"M0280",
"N0099"
] | 122 | 2023-01-27T15:35:20 | [
"1",
"1",
"2",
"2",
"3",
"4",
"5",
"6",
"7",
"8",
"11",
"12",
"15",
"16",
"19",
"22",
"25",
"28",
"31",
"34",
"40",
"43",
"49",
"52",
"58",
"64",
"70",
"76",
"82",
"88",
"98",
"104",
"114",
"120",
"130",
"140",
"150",
"160",
"170",
"180",
"195",
"205",
"220",
"230",
"245",
"260",
"275",
"290",
"305",
"320",
"341",
"356",
"377",
"392",
"413",
"434",
"455",
"476",
"497",
"518",
"546"
] | Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents. | [
"A000008",
"A001299",
"A001300",
"A025810",
"A169718"
] | N. J. A. Sloane | [
"nonn",
"easy",
"nice"
] | 0 | 5 | oeisdata/seq/A000/A000008.seq | %I A000008 M0280 N0099 #122 Jan 27 2023 15:35:20
%S A000008 1,1,2,2,3,4,5,6,7,8,11,12,15,16,19,22,25,28,31,34,40,43,49,52,58,64,
%T A000008 70,76,82,88,98,104,114,120,130,140,150,160,170,180,195,205,220,230,
%U A000008 245,260,275,290,305,320,341,356,377,392,413,434,455,476,497,518,546
%N A000008 Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
%C A000008 Number of partitions of n into parts 1, 2, 5, and 10.
%C A000008 There is a unique solution to this puzzle: "There are a prime number of ways that I can make change for n cents using coins of 1, 2, 5, 10 cents; but a semiprime number of ways that I can make change for n-1 cents and for n+1 cents." There is a unique solution to this related puzzle: "There are a prime number of ways that I can make change for n cents using coins of 1, 2, 5, 10 cents; but a 3-almost prime number of ways that I can make change for n-1 cents and for n+1 cents." - _Jonathan Vos Post_, Aug 26 2005
%D A000008 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
%D A000008 G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.
%D A000008 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 152.
%D A000008 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000008 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000008 William Boyles, <a href="/A000008/b000008.txt">Table of n, a(n) for n = 0..10000</a> (terms 0...1000 from T. D. Noe)
%H A000008 Henry Bottomley, <a href="/A000008/a000008.gif">Initial terms of A000008, A001301, A001302, A001312, A001313</a>
%H A000008 X. Gourdon and B. Salvy, <a href="http://dx.doi.org/10.1016/0012-365X(95)00133-H">Effective asymptotics of linear recurrences with rational coefficients</a>, Discrete Mathematics, vol. 153, no. 1-3, 1996, pages 145-163.
%H A000008 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=174">Encyclopedia of Combinatorial Structures 174</a>
%H A000008 Gerhard Kirchner, <a href="/A187243/a187243_1.pdf">Derivation of formulas</a>
%H A000008 <a href="/index/Mag#change">Index entries for sequences related to making change.</a>
%H A000008 <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,1,-1,-1,1,0,1,-1,-1,1,0,-1,1,1,-1).
%F A000008 G.f.: 1 / ((1 - x) * (1 - x^2) * (1 - x^5) * (1 - x^10)). - _Michael Somos_, Nov 17 1999
%F A000008 a(n) - a(n-1) = A025810(n). - _Michael Somos_, Dec 15 2002
%F A000008 a(n) = a(n-2) + a(n-5) - a(n-7) + a(n-10) - a(n-12) - a(n-15) + a(n-17) + 1. - _Michael Somos_, Apr 01 2003
%F A000008 a(n) = -a(-18-n). - _Michael Somos_, Apr 01 2003
%F A000008 a(n) = (q+1)*(h(n) - q*(3n-10q+7)/6) with q = floor(n/10) and h(n) = A000115(n) = round((n+4)^2/20). See link "Derivation of formulas". - _Gerhard Kirchner_, Feb 10 2017
%e A000008 G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + 8*x^9 + 11*x^10 + ...
%p A000008 M:= Matrix(18, (i,j)-> if(i=j-1 and i<17) or (j=1 and member(i, [2,5,10,17,18])) or (i=18 and j=18) then 1 elif j=1 and member(i, [7,12,15]) then -1 else 0 fi); a:= n-> (M^(n+1))[18,1]; seq(a(n), n=0..51); # _Alois P. Heinz_, Jul 25 2008
%p A000008 # second Maple program:
%p A000008 a:= proc(n) local m, r; m := iquo(n, 10, 'r'); r:= r+1; ([23, 26, 35, 38, 47, 56, 65, 74, 83, 92][r]+ (3*r+ 24+ 10*m) *m) *m/6+ [1, 1, 2, 2, 3, 4, 5, 6, 7, 8][r] end: seq(a(n), n=0..100); # _Alois P. Heinz_, Oct 05 2008
%t A000008 a[ n_] := SeriesCoefficient[ 1 / ((1 - x)(1 - x^2)(1 - x^5)(1 - x^10)), {x, 0, n}]
%t A000008 a[n_, d_] := SeriesCoefficient[1/(Times@@Map[(1-x^#)&, d]), {x, 0, n}] (* general case for any set of denominations represented as a list d of coin values in cents *)
%t A000008 Table[Length[FrobeniusSolve[{1,2,5,10},n]],{n,0,70}] (* _Harvey P. Dale_, Apr 02 2012 *)
%t A000008 LinearRecurrence[{1, 1, -1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1}, {1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16, 19, 22, 25, 28}, 100] (* _Vincenzo Librandi_, Feb 10 2016 *)
%t A000008 a[ n_] := Quotient[ With[{r = Mod[n, 10, 1]}, n^3 + 27 n^2 + (191 + 3 {4, 13, 0, 5, 8, 9, 8, 5, 0, 13}[[r]]) n + 25], 600] + 1; (* _Michael Somos_, Mar 06 2018 *)
%t A000008 Table[Length@IntegerPartitions[n,All,{1,2,5,10}],{n,0,70}] (* _Giorgos Kalogeropoulos_, May 07 2019 *)
%o A000008 (PARI) {a(n) = if( n<-17, -a(-18-n), if( n<0, 0, polcoeff( 1 / ((1 - x) * (1 - x^2) * (1 - x^5) * (1 - x^10)) + x * O(x^n), n)))}; /* _Michael Somos_, Apr 01 2003 */
%o A000008 (PARI) Vec( 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)) + O(x^66) ) \\ _Joerg Arndt_, Oct 02 2013
%o A000008 (PARI) {a(n) = my(r = (n-1)%10 + 1); (n^3 + 27*n^2 + (191 + 3*[4, 13, 0, 5, 8, 9, 8, 5, 0, 13][r])*n + 25)\600 + 1}; /* _Michael Somos_, Mar 06 2018 */
%o A000008 (Maxima) a(n):=floor(((n+17)*(2*n^2+20*n+81)+15*(n+1)*(-1)^n+120*((floor(n/5)+1)*((1+(-1)^mod(n,5))/2-floor(((mod(n,5))^2)/8))))/1200); /* _Tani Akinari_, Jun 21 2013 */
%o A000008 (Haskell)
%o A000008 a000008 = p [1,2,5,10] where
%o A000008 p _ 0 = 1
%o A000008 p [] _ = 0
%o A000008 p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o A000008 -- _Reinhard Zumkeller_, Dec 15 2013
%o A000008 (Magma) [#RestrictedPartitions(n,{1,2,5,10}):n in [0..60]]; // _Marius A. Burtea_, May 07 2019
%Y A000008 Cf. A001299, A025810.
%Y A000008 Cf. A001300, A169718.
%K A000008 nonn,easy,nice
%O A000008 0,3
%A A000008 _N. J. A. Sloane_
| f77419dc9ad5a3a3930268e86c5fbef4 |
A000009 | [
"M0281",
"N0100"
] | 529 | 2025-02-16T08:32:18 | [
"1",
"1",
"1",
"2",
"2",
"3",
"4",
"5",
"6",
"8",
"10",
"12",
"15",
"18",
"22",
"27",
"32",
"38",
"46",
"54",
"64",
"76",
"89",
"104",
"122",
"142",
"165",
"192",
"222",
"256",
"296",
"340",
"390",
"448",
"512",
"585",
"668",
"760",
"864",
"982",
"1113",
"1260",
"1426",
"1610",
"1816",
"2048",
"2304",
"2590",
"2910",
"3264",
"3658",
"4097",
"4582",
"5120",
"5718",
"6378"
] | Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts. | [
"A000009",
"A000041",
"A000700",
"A000726",
"A001318",
"A001935",
"A003724",
"A004111",
"A007837",
"A010054",
"A010815",
"A015723",
"A035294",
"A035363",
"A035959",
"A035985",
"A052839",
"A053632",
"A057077",
"A067659",
"A067661",
"A068049",
"A078408",
"A081360",
"A088670",
"A089806",
"A091602",
"A097306",
"A104502",
"A109950",
"A109968",
"A118457",
"A118459",
"A132312",
"A146061",
"A167377",
"A219601",
"A230957",
"A237515",
"A261775",
"A261776",
"A328545",
"A328546"
] | N. J. A. Sloane | [
"nonn",
"core",
"easy",
"nice"
] | 0 | 5 | oeisdata/seq/A000/A000009.seq | %I A000009 M0281 N0100 #529 Feb 16 2025 08:32:18
%S A000009 1,1,1,2,2,3,4,5,6,8,10,12,15,18,22,27,32,38,46,54,64,76,89,104,122,
%T A000009 142,165,192,222,256,296,340,390,448,512,585,668,760,864,982,1113,
%U A000009 1260,1426,1610,1816,2048,2304,2590,2910,3264,3658,4097,4582,5120,5718,6378
%N A000009 Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts.
%C A000009 Partitions into distinct parts are sometimes called "strict partitions".
%C A000009 The number of different ways to run up a staircase with m steps, taking steps of odd sizes (or taking steps of distinct sizes), where the order is not relevant and there is no other restriction on the number or the size of each step taken is the coefficient of x^m.
%C A000009 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A000009 The result that number of partitions of n into distinct parts = number of partitions of n into odd parts is due to Euler.
%C A000009 Bijection: given n = L1* 1 + L2*3 + L3*5 + L7*7 + ..., a partition into odd parts, write each Li in binary, Li = 2^a1 + 2^a2 + 2^a3 + ... where the aj's are all different, then expand n = (2^a1 * 1 + ...)*1 + ... by removing the brackets and we get a partition into distinct parts. For the reverse operation, just keep splitting any even number into halves until no evens remain.
%C A000009 Euler transform of period 2 sequence [1,0,1,0,...]. - _Michael Somos_, Dec 16 2002
%C A000009 Number of different partial sums 1+[1,2]+[1,3]+[1,4]+..., where [1,x] indicates a choice. E.g., a(6)=4, as we can write 1+1+1+1+1+1, 1+2+3, 1+2+1+1+1, 1+1+3+1. - _Jon Perry_, Dec 31 2003
%C A000009 a(n) is the sum of the number of partitions of x_j into at most j parts, where j is the index for the j-th triangular number and n-T(j)=x_j. For example; a(12)=partitions into <= 4 parts of 12-T(4)=2 + partitions into <= 3 parts of 12-T(3)=6 + partitions into <= 2 parts of 12-T(2)=9 + partitions into 1 part of 12-T(1)=11 = (2)(11) + (6)(51)(42)(411)(33)(321)(222) + (9)(81)(72)(63)(54)+(11) = 2+7+5+1 = 15. - _Jon Perry_, Jan 13 2004
%C A000009 Number of partitions of n where if k is the largest part, all parts 1..k are present. - _Jon Perry_, Sep 21 2005
%C A000009 Jack Grahl and Franklin T. Adams-Watters prove this claim of Jon Perry's by observing that the Ferrers dual of a "gapless" partition is guaranteed to have distinct parts; since the Ferrers dual is an involution, this establishes a bijection between the two sets of partitions. - _Allan C. Wechsler_, Sep 28 2021
%C A000009 The number of connected threshold graphs having n edges. - Michael D. Barrus (mbarrus2(AT)uiuc.edu), Jul 12 2007
%C A000009 Starting with offset 1 = row sums of triangle A146061 and the INVERT transform of A000700 starting: (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 3, -3, 4, -5, ...). - _Gary W. Adamson_, Oct 26 2008
%C A000009 Number of partitions of n in which the largest part occurs an odd number of times and all other parts occur an even number of times. (Such partitions are the duals of the partitions with odd parts.) - _David Wasserman_, Mar 04 2009
%C A000009 Equals A035363 convolved with A010054. The convolution square of A000009 = A022567 = A000041 convolved with A010054. A000041 = A000009 convolved with A035363. - _Gary W. Adamson_, Jun 11 2009
%C A000009 Considering all partitions of n into distinct parts: there are A140207(n) partitions of maximal size which is A003056(n), and A051162(n) is the greatest number occurring in these partitions. - _Reinhard Zumkeller_, Jun 13 2009
%C A000009 Equals left border of triangle A091602 starting with offset 1. - _Gary W. Adamson_, Mar 13 2010
%C A000009 Number of symmetric unimodal compositions of n+1 where the maximal part appears once. Also number of symmetric unimodal compositions of n where the maximal part appears an odd number of times. - _Joerg Arndt_, Jun 11 2013
%C A000009 Because for these partitions the exponents of the parts 1, 2, ... are either 0 or 1 (j^0 meaning that part j is absent) one could call these partitions also 'fermionic partitions'. The parts are the levels, that is the positive integers, and the occupation number is either 0 or 1 (like Pauli's exclusion principle). The 'fermionic states' are denoted by these partitions of n. - _Wolfdieter Lang_, May 14 2014
%C A000009 The set of partitions containing only odd parts forms a monoid under the product described in comments to A047993. - _Richard Locke Peterson_, Aug 16 2018
%C A000009 Ewell (1973) gives a number of recurrences. - _N. J. A. Sloane_, Jan 14 2020
%C A000009 a(n) equals the number of permutations p of the set {1,2,...,n+1}, written in one line notation as p = p_1p_2...p_(n+1), satisfying p_(i+1) - p_i <= 1 for 1 <= i <= n, (i.e., those permutations that, when read from left to right, never increase by more than 1) whose major index maj(p) := Sum_{p_i > p_(i+1)} i equals n. For example, of the 16 permutations on 5 letters satisfying p_(i+1) - p_i <= 1, 1 <= i <= 4, there are exactly two permutations whose major index is 4, namely, 5 3 4 1 2 and 2 3 4 5 1. Hence a(4) = 2. See the Bala link in A007318 for a proof. - _Peter Bala_, Mar 30 2022
%C A000009 Conjecture: Each positive integer n can be written as a_1 + ... + a_k, where a_1,...,a_k are strict partition numbers (i.e., terms of the current sequence) with no one dividing another. This has been verified for n = 1..1350. - _Zhi-Wei Sun_, Apr 14 2023
%C A000009 Conjecture: For each integer n > 7, a(n) divides none of p(n), p(n) - 1 and p(n) + 1, where p(n) is the number of partitions of n given by A000041. This has been verified for n up to 10^5. - _Zhi-Wei Sun_, May 20 2023 [Verified for n <= 2*10^6. - _Vaclav Kotesovec_, May 23 2023]
%C A000009 The g.f. Product_{k >= 0} 1 + x^k = Product_{k >= 0} 1 - x^k + 2*x^k == Product_{k >= 0} 1 - x^k == Sum_{k in Z} (-1)^k*x^(k*(3*k-1)/2) (mod 2) by Euler's pentagonal number theorem. It follows that a(n) is odd iff n = k*(3*k - 1)/2 for some integer k, i.e., iff n is a generalized pentagonal number A001318. - _Peter Bala_, Jan 07 2025
%D A000009 Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
%D A000009 Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.
%D A000009 George E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.
%D A000009 George E. Andrews, Number Theory, Dover Publications, 1994, Theorem 12-3, pp. 154-5, and (13-1-1) p. 163.
%D A000009 Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; see p. 196.
%D A000009 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, Problem 18.
%D A000009 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 99.
%D A000009 William Dunham, The Mathematical Universe, pp. 57-62, J. Wiley, 1994.
%D A000009 Leonhard Euler, De partitione numerorum, Novi commentarii academiae scientiarum Petropolitanae 3 (1750/1), 1753, reprinted in: Commentationes Arithmeticae. (Opera Omnia. Series Prima: Opera Mathematica, Volumen Secundum), 1915, Lipsiae et Berolini, 254-294.
%D A000009 Ian P. Goulden and David M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.1).
%D A000009 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 277, Theorems 344, 346.
%D A000009 Carlos J. Moreno and Samuel S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 253.
%D A000009 Srinivasa Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. See Table V on page 309.
%D A000009 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000009 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000009 Reinhard Zumkeller, <a href="/A000009/b000009.txt">Table of n, a(n) for n = 0..5000</a> (first 2000 terms from N. J. A. Sloane)
%H A000009 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp. 348-350.
%H A000009 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 836.
%H A000009 Francesca Aicardi, <a href="https://doi.org/10.1007/s11853-011-0045-z">Matricial formulae for partitions</a>, Functional Analysis and Other Mathematics, Vol. 3, No. 2 (2011), pp. 127-133; <a href="http://arxiv.org/abs/0806.1273">arXiv preprint</a>, arXiv:0806.1273 [math.NT], 2008.
%H A000009 George E. Andrews, <a href="http://dx.doi.org/10.1090/S0273-0979-07-01180-9">Euler's "De Partitio Numerorum"</a>, Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
%H A000009 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/320.pdf">The Bhargava-Adiga Summation and Partitions</a>, 2016. See page 4 equation (2.2).
%H A000009 Brennan Benfield and Arindam Roy, <a href="https://arxiv.org/abs/2404.03153">Log-concavity And The Multiplicative Properties of Restricted Partition Functions</a>, arXiv:2404.03153 [math.NT], 2024.
%H A000009 Helena Bergold, Lukas Egeling, and Hung. P. Hoang, <a href="https://arxiv.org/abs/2411.19208">Signotopes with few plus signs</a>, arXiv:2411.19208 [math.CO], 2024. See p. 14.
%H A000009 Andreas B. G. Blobel, <a href="https://arxiv.org/abs/1904.07808">An Asymptotic Form of the Generating Function Prod_{k=1,oo} (1+x^k/k)</a>, arXiv:1904.07808 [math.CO], 2019.
%H A000009 Alexander Bors, Michael Giudici, and Cheryl E. Praeger, <a href="https://arxiv.org/abs/1910.12570">Documentation for the GAP code file OrbOrd.txt</a>, arXiv:1910.12570 [math.GR], 2019.
%H A000009 Henry Bottomley, <a href="/A000009/a000009.gif">Illustration for A000009, A000041, A047967</a>.
%H A000009 Andrés Eduardo Caicedo and Brittany Shelton, <a href="https://doi.org/10.1080/0025570X.2018.1403233">Of puzzles and partitions: Introducing Partiti</a>, Mathematics Magazine, Vol. 91, No. 1 (2018), pp. 20-23; <a href="https://arxiv.org/abs/1710.04495">arXiv preprint</a>, arXiv:1710.04495 [math.HO], 2017.
%H A000009 Huantian Cao, <a href="http://cobweb.cs.uga.edu/~rwr/STUDENTS/hcao.html">AutoGF: An Automated System to Calculate Coefficients of Generating Functions</a>, thesis, 2002.
%H A000009 Huantian Cao, <a href="/A000009/a000009.pdf">AutoGF: An Automated System to Calculate Coefficients of Generating Functions</a>, thesis, 2002. [Local copy, with permission]
%H A000009 H. B. C. Darling, Collected Papers of Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper36/page34.htm">Table for q(n); n=1 through 100</a>.
%H A000009 Alejandro Erickson and Mark Schurch, <a href="https://doi.org/10.1016/j.jda.2012.04.002">Monomer-dimer tatami tilings of square regions</a>, Journal of Discrete Algorithms, Vol. 16 (2012), pp. 258-269; <a href="http://arxiv.org/abs/1110.5103">arXiv preprint</a>, arXiv:1110.5103 [math.CO], 2011.
%H A000009 John A. Ewell, <a href="https://doi.org/10.1016/0097-3165(73)90070-8">Partition recurrences</a>, J. Comb. Theory A, Vol. 14, 125-127, 1973.
%H A000009 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, Cambridge University Press, 2009; see pages 48 and 499.
%H A000009 Evangelos Georgiadis, <a href="https://web.archive.org/web/20131109101330/http://web.mit.edu/egeorg/Public/Partitions/PartitionsQ.pdf">Computing Partition Numbers q(n)</a>, Technical Report, February (2009).
%H A000009 Benjamin Hackl, <a href="https://www.youtube.com/watch?v=M4TmnYxS4gk">5 + 5 + 1 + 1 + 1 = 10 + 2 + 1, and why there is more to it than you think.</a>, YouTube video, 2022.
%H A000009 Cristiano Husu, <a href="https://doi.org/10.1016/j.jnt.2018.05.005">The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts</a>, Journal of Number Theory, Vol. 193 (2018), pp. 171-188; <a href="https://arxiv.org/abs/1804.09883">arXiv preprint</a>, arXiv:1804.09883 [math.NT], 2018.
%H A000009 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=108">Encyclopedia of Combinatorial Structures 108</a>.
%H A000009 Martin Klazar, <a href="http://arxiv.org/abs/1808.08449">What is an answer? - remarks, results and problems on PIO formulas in combinatorial enumeration, part I</a>, arXiv:1808.08449 [math.CO], 2018.
%H A000009 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.
%H A000009 Vaclav Kotesovec, <a href="http://mathematica.stackexchange.com/questions/130741/getting-wrong-limit-with-bessel">Getting wrong limit with Bessel</a>, Mathematica Stack Exchange, Nov 09 2016.
%H A000009 Alain Lascoux, <a href="http://dx.doi.org/10.1016/j.disc.2002.02.001">Sylvester's bijection between strict and odd partitions</a>, Discrete Math., Vol. 277, No. 1-3 (2004), pp. 275-278.
%H A000009 Jeremy Lovejoy, <a href="https://doi.org/10.1112/S0024609302001492">The number of partitions into distinct parts modulo powers of 5</a>, Bulletin of the London Mathematical Society, Vol. 35, No. 1 (2003), pp. 41-46; <a href="http://lovejoy.perso.math.cnrs.fr/5powersQ.pdf">alternative link</a>.
%H A000009 James Mc Laughlin, Andrew V. Sills, and Peter Zimmer, <a href="https://doi.org/10.37236/36">Rogers-Ramanujan-Slater Type Identities</a>, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008; see also <a href="https://arxiv.org/abs/1901.00946">arXiv version</a>, arXiv:1901.00946 [math.NT], 2019.
%H A000009 Günter Meinardus, <a href="https://eudml.org/doc/169463">Über Partitionen mit Differenzenbedingungen</a>, Mathematische Zeitschrift (1954/55), Volume 61, page 289-302.
%H A000009 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function q(n).
%H A000009 Mircea Merca, <a href="https://doi.org/10.1007/s11139-016-9856-3">The Lambert series factorization theorem</a>, The Ramanujan Journal, Vol. 44, No. 2 (2017), pp. 417-435; <a href="https://www.researchgate.net/publication/312324402">alternative link</a>.
%H A000009 István Mező, <a href="https://arxiv.org/abs/1106.2703">Several special values of Jacobi theta functions</a> arXiv:1106.2703v3 [math.CA], 2011-2013.
%H A000009 Donald J. Newman, <a href="http://dx.doi.org/10.1007/978-1-4613-8214-0">A Problem Seminar</a>, pp. 18;93;102-3 Prob. 93 Springer-Verlag NY 1982.
%H A000009 Hieu D. Nguyen and Douglas Taggart, <a href="http://citeseerx.ist.psu.edu/pdf/8f2f36f22878c984775ed04368b8893879b99458">Mining the OEIS: Ten Experimental Conjectures</a>, 2013. Mentions this sequence.
%H A000009 Kimeu Arphaxad Ngwava, Nick Gill, and Ian Short, <a href="https://arxiv.org/abs/2005.13869">Nilpotent covers of symmetric groups</a>, arXiv:2005.13869 [math.GR], 2020.
%H A000009 Matthew Parker, <a href="https://oeis.org/A000009/a000009_100K.7z">The first 100K terms (7-Zip compressed file)</a>.
%H A000009 Marko Riedel, <a href="https://math.stackexchange.com/questions/3063971/">Proof of recurrence by V. Jovovic</a>.
%H A000009 Ed Sandifer, How Euler Did It, <a href="http://eulerarchive.maa.org/hedi/HEDI-2005-10.pdf">Philip Naude's problem</a>.
%H A000009 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A000009 Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/444761">A representation problem involving strict partition numbers</a>, Question 444761 at MathOverflow, April 14, 2023.
%H A000009 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>, <a href="https://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q</a>, <a href="https://mathworld.wolfram.com/PartitionFunctionb.html">Partition Function bk</a>, <a href="https://mathworld.wolfram.com/EulerIdentity.html">Euler Identity</a>, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.
%H A000009 Wikipedia, <a href="https://en.wikipedia.org/wiki/Glaisher%27s_theorem">Glaisher's Theorem</a>.
%H A000009 Wolfram Research, <a href="http://functions.wolfram.com/IntegerFunctions/PartitionsQ/11">Generating functions for q(n)</a>.
%H A000009 Mingjia Yang and Doron Zeilberger, <a href="https://arxiv.org/abs/1910.08989">Systematic Counting of Restricted Partitions</a>, arXiv:1910.08989 [math.CO], 2019.
%H A000009 Michael P. Zaletel and Roger S. K. Mong, <a href="https://doi.org/10.1103/PhysRevB.86.245305">Exact matrix product states for quantum Hall wave functions</a>, Physical Review B, Vol. 86, No. 24 (2012), 245305; <a href="http://arxiv.org/abs/1208.4862">arXiv preprint</a>, arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012
%H A000009 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F A000009 G.f.: Product_{m>=1} (1 + x^m) = 1/Product_{m>=0} (1-x^(2m+1)) = Sum_{k>=0} Product_{i=1..k} x^i/(1-x^i) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1-x^k).
%F A000009 G.f.: Sum_{n>=0} x^n*Product_{k=1..n-1} (1+x^k) = 1 + Sum_{n>=1} x^n*Product_{k>=n+1} (1+x^k). - _Joerg Arndt_, Jan 29 2011
%F A000009 Product_{k>=1} (1+x^(2k)) = Sum_{k>=0} x^(k*(k+1))/Product_{i=1..k} (1-x^(2i)) - Euler (Hardy and Wright, Theorem 346).
%F A000009 Asymptotics: a(n) ~ exp(Pi l_n / sqrt(3)) / ( 4 3^(1/4) l_n^(3/2) ) where l_n = (n-1/24)^(1/2) (Ayoub).
%F A000009 For n > 1, a(n) = (1/n)*Sum_{k=1..n} b(k)*a(n-k), with a(0)=1, b(n) = A000593(n) = sum of odd divisors of n; cf. A000700. - _Vladeta Jovovic_, Jan 21 2002
%F A000009 a(n) = t(n, 0), t as defined in A079211.
%F A000009 a(n) = Sum_{k=0..n-1} A117195(n,k) = A117192(n) + A117193(n) for n>0. - _Reinhard Zumkeller_, Mar 03 2006
%F A000009 a(n) = A026837(n) + A026838(n) = A118301(n) + A118302(n); a(A001318(n)) = A051044(n); a(A090864(n)) = A118303(n). - _Reinhard Zumkeller_, Apr 22 2006
%F A000009 Expansion of 1 / chi(-x) = chi(x) / chi(-x^2) = f(-x) / phi(x) = f(x) / phi(-x^2) = psi(x) / f(-x^2) = f(-x^2) / f(-x) = f(-x^4) / psi(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_, Mar 12 2011
%F A000009 G.f. is a period 1 Fourier series which satisfies f(-1 / (1152 t)) = 2^(-1/2) / f(t) where q = exp(2 Pi i t). - _Michael Somos_, Aug 16 2007
%F A000009 Expansion of q^(-1/24) * eta(q^2) / eta(q) in powers of q.
%F A000009 Expansion of q^(-1/24) 2^(-1/2) f2(t) in powers of q = exp(2 Pi i t) where f2() is a Weber function. - _Michael Somos_, Oct 18 2007
%F A000009 Given g.f. A(x), then B(x) = x * A(x^3)^8 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = v - u^2 + 16*u*v^2 . - _Michael Somos_, May 31 2005
%F A000009 Given g.f. A(x), then B(x) = x * A(x^8)^3 satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (u^3 - v) * (u + v^3) - 9 * u^3 * v^3. - _Michael Somos_, Mar 25 2008
%F A000009 From Evangelos Georgiadis, Andrew V. Sutherland, Kiran S. Kedlaya (egeorg(AT)mit.edu), Mar 03 2009: (Start)
%F A000009 a(0)=1; a(n) = 2*(Sum_{k=1..floor(sqrt(n))} (-1)^(k+1) a(n-k^2)) + sigma(n) where sigma(n) = (-1)^j if (n=(j*(3*j+1))/2 OR n=(j*(3*j-1))/2) otherwise sigma(n)=0 (simpler: sigma = A010815). (End)
%F A000009 From _Gary W. Adamson_, Jun 13 2009: (Start)
%F A000009 The product g.f. = (1/(1-x))*(1/(1-x^3))*(1/(1-x^5))*...; = (1,1,1,...)*
%F A000009 (1,0,0,1,0,0,1,0,0,1,...)*(1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,...) * ...; =
%F A000009 a*b*c*... where a, a*b, a*b*c, ... converge to A000009:
%F A000009 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, ... = a*b
%F A000009 1, 1, 1, 2, 2, 3, 4, 4, 5, 6, ... = a*b*c
%F A000009 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, ... = a*b*c*d
%F A000009 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, ... = a*b*c*d*e
%F A000009 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, ... = a*b*c*d*e*f
%F A000009 ... (cf. analogous example in A000041). (End)
%F A000009 a(A004526(n)) = A172033(n). - _Reinhard Zumkeller_, Jan 23 2010
%F A000009 a(n) = P(n) - P(n-2) - P(n-4) + P(n-10) + P(n-14) + ... + (-1)^m P(n-2p_m) + ..., where P(n) is the partition function (A000041) and p_m = m(3m-1)/2 is the m-th generalized pentagonal number (A001318). - _Jerome Malenfant_, Feb 16 2011
%F A000009 a(n) = A054242(n,0) = A201377(n,0). - _Reinhard Zumkeller_, Dec 02 2011
%F A000009 G.f.: 1/2 (-1; x)_{inf} where (a; q)_k is the q-Pochhammer symbol. - _Vladimir Reshetnikov_, Apr 24 2013
%F A000009 More precise asymptotics: a(n) ~ exp(Pi*sqrt((n-1/24)/3)) / (4*3^(1/4)*(n-1/24)^(3/4)) * (1 + (Pi^2-27)/(24*Pi*sqrt(3*(n-1/24))) + (Pi^4-270*Pi^2-1215)/(3456*Pi^2*(n-1/24))). - _Vaclav Kotesovec_, Nov 30 2015
%F A000009 a(n) = A067661(n) + A067659(n). _Wolfdieter Lang_, Jan 18 2016
%F A000009 From _Vaclav Kotesovec_, May 29 2016: (Start)
%F A000009 a(n) ~ exp(Pi*sqrt(n/3))/(4*3^(1/4)*n^(3/4)) * (1 + (Pi/(48*sqrt(3)) - (3*sqrt(3))/(8*Pi))/sqrt(n) + (Pi^2/13824 - 5/128 - 45/(128*Pi^2))/n).
%F A000009 a(n) ~ exp(Pi*sqrt(n/3) + (Pi/(48*sqrt(3)) - 3*sqrt(3)/(8*Pi))/sqrt(n) - (1/32 + 9/(16*Pi^2))/n) / (4*3^(1/4)*n^(3/4)).
%F A000009 (End)
%F A000009 a(n) = A089806(n)*A010815(floor(n/2)) + a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) + ... + A057077(m-1)*a(n-A001318(m)) + ..., where n > A001318(m). - _Gevorg Hmayakyan_, Jul 07 2016
%F A000009 a(n) ~ Pi*BesselI(1, Pi*sqrt((n+1/24)/3)) / sqrt(24*n+1). - _Vaclav Kotesovec_, Nov 08 2016
%F A000009 a(n) = A000041(n) - A047967(n). - _R. J. Mathar_, Nov 20 2017
%F A000009 Sum_{n>=1} 1/a(n) = A237515. - _Amiram Eldar_, Nov 15 2020
%F A000009 From _Peter Bala_, Jan 15 2021: (Start)
%F A000009 G.f.: (1 + x)*Sum_{n >= 0} x^(n*(n+3)/2)/Product_{k = 1..n} (1 - x^k) =
%F A000009 (1 + x)*(1 + x^2)*Sum_{n >= 0} x^(n*(n+5)/2)/Product_{k = 1..n} (1 - x^k) = (1 + x)*(1 + x^2)*(1 + x^3)*Sum_{n >= 0} x^(n*(n+7)/2)/Product_{k = 1..n} (1 - x^k) = ....
%F A000009 G.f.: (1/2)*Sum_{n >= 0} x^(n*(n-1)/2)/Product_{k = 1..n} (1 - x^k) =
%F A000009 (1/2)*(1/(1 + x))*Sum_{n >= 0} x^((n-1)*(n-2)/2)/Product_{k = 1..n} (1 - x^k) = (1/2)*(1/((1 + x)*(1 + x^2)))*Sum_{n >= 0} x^((n-2)*(n-3)/2)/Product_{k = 1..n} (1 - x^k) = ....
%F A000009 G.f.: Sum_{n >= 0} x^n/Product_{k = 1..n} (1 - x^(2*k)) = (1/(1 - x)) * Sum_{n >= 0} x^(3*n)/Product_{k = 1..n} (1 - x^(2*k)) = (1/((1 - x)*(1 - x^3))) * Sum_{n >= 0} x^(5*n)/Product_{k = 1..n} (1 - x^(2*k)) = (1/((1 - x)*(1 - x^3)*(1 - x^5))) * Sum_{n >= 0} x^(7*n)/Product_{k = 1..n} (1 - x^(2*k)) = .... (End)
%F A000009 From _Peter Bala_, Feb 02 2021: (Start)
%F A000009 G.f.: A(x) = Sum_{n >= 0} x^(n*(2*n-1))/Product_{k = 1..2*n} (1 - x^k). (Set z = x and q = x^2 in Mc Laughlin et al. (2019 ArXiv version), Section 1.3, Identity 7.)
%F A000009 Similarly, A(x) = Sum_{n >= 0} x^(n*(2*n+1))/Product_{k = 1..2*n+1} (1 - x^k). (End)
%F A000009 a(n) = A001227(n) + A238005(n) + A238006(n). - _R. J. Mathar_, Sep 08 2021
%F A000009 G.f.: A(x) = exp ( Sum_{n >= 1} x^n/(n*(1 - x^(2*n))) ) = exp ( Sum_{n >= 1} (-1)^(n+1)*x^n/(n*(1 - x^n)) ). - _Peter Bala_, Dec 23 2021
%F A000009 Sum_{n>=0} a(n)/exp(Pi*n) = exp(Pi/24)/2^(1/8) = A292820. - _Simon Plouffe_, May 12 2023 [Proof: Sum_{n>=0} a(n)/exp(Pi*n) = phi(exp(-2*Pi)) / phi(exp(-Pi)), where phi(q) is the Euler modular function. We have phi(exp(-2*Pi)) = exp(Pi/12) * Gamma(1/4) / (2 * Pi^(3/4)) and phi(exp(-Pi)) = exp(Pi/24) * Gamma(1/4) / (2^(7/8) * Pi^(3/4)), see formulas (14) and (13) in I. Mező, 2013. - _Vaclav Kotesovec_, May 12 2023]
%F A000009 a(2*n) = Sum_{j=1..n} p(n+j, 2*j) and a(2*n+1) = Sum_{j=1..n+1} p(n+j,2*j-1), where p(n, s) is the number of partitions of n having exactly s parts. - _Gregory L. Simay_, Aug 30 2023
%e A000009 G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 8*x^9 + ...
%e A000009 G.f. = q + q^25 + q^49 + 2*q^73 + 2*q^97 + 3*q^121 + 4*q^145 + 5*q^169 + ...
%e A000009 The partitions of n into distinct parts (see A118457) for small n are:
%e A000009 1: 1
%e A000009 2: 2
%e A000009 3: 3, 21
%e A000009 4: 4, 31
%e A000009 5: 5, 41, 32
%e A000009 6: 6, 51, 42, 321
%e A000009 7: 7, 61, 52, 43, 421
%e A000009 8: 8, 71, 62, 53, 521, 431
%e A000009 ...
%e A000009 From _Reinhard Zumkeller_, Jun 13 2009: (Start)
%e A000009 a(8)=6, A140207(8)=#{5+2+1,4+3+1}=2, A003056(8)=3, A051162(8)=5;
%e A000009 a(9)=8, A140207(9)=#{6+2+1,5+3+1,4+3+2}=3, A003056(9)=3, A051162(9)=6;
%e A000009 a(10)=10, A140207(10)=#{4+3+2+1}=1, A003056(10)=4, A051162(10)=4. (End)
%p A000009 N := 100; t1 := series(mul(1+x^k,k=1..N),x,N); A000009 := proc(n) coeff(t1,x,n); end;
%p A000009 spec := [ P, {P=PowerSet(N), N=Sequence(Z,card>=1)} ]: [ seq(combstruct[count](spec, size=n), n=0..58) ];
%p A000009 spec := [ P, {P=PowerSet(N), N=Sequence(Z,card>=1)} ]: combstruct[allstructs](spec, size=10); # to get the actual partitions for n=10
%p A000009 A000009 := proc(n)
%p A000009 local x,m;
%p A000009 product(1+x^m,m=1..n+1) ;
%p A000009 expand(%) ;
%p A000009 coeff(%,x,n) ;
%p A000009 end proc: # _R. J. Mathar_, Jun 18 2016
%p A000009 # Alternatively:
%p A000009 simplify(expand(QDifferenceEquations:-QPochhammer(-1,x,99)/2,x)):
%p A000009 seq(coeff(%,x,n), n=0..55); # _Peter Luschny_, Nov 17 2016
%t A000009 PartitionsQ[Range[0, 60]] (* _Harvey Dale_, Jul 27 2009 *)
%t A000009 a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, n}], {x, 0, n}]; (* _Michael Somos_, Jul 06 2011 *)
%t A000009 a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^k, {k, 1, n, 2}], {x, 0, n}]; (* _Michael Somos_, Jul 06 2011 *)
%t A000009 a[ n_] := With[ {t = Log[q] / (2 Pi I)}, SeriesCoefficient[ q^(-1/24) DedekindEta[2 t] / DedekindEta[ t], {q, 0, n}]]; (* _Michael Somos_, Jul 06 2011 *)
%t A000009 a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ x, x^2], {x, 0, n}]; (* _Michael Somos_, May 24 2013 *)
%t A000009 a[ n_] := SeriesCoefficient[ Series[ QHypergeometricPFQ[ {q}, {q x}, q, - q x], {q, 0, n}] /. x -> 1, {q, 0, n}]; (* _Michael Somos_, Mar 04 2014 *)
%t A000009 a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[{}, {}, q, -1] / 2, {q, 0, n}]; (* _Michael Somos_, Mar 04 2014 *)
%t A000009 nmax = 60; CoefficientList[Series[Exp[Sum[(-1)^(k+1)/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 25 2015 *)
%t A000009 nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Jan 14 2017 *)
%o A000009 (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n))}; /* _Michael Somos_, Nov 17 1999 */
%o A000009 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A), n))};
%o A000009 (PARI) {a(n) = my(c); forpart(p=n, if( n<1 || p[1]<2, c++; for(i=1, #p-1, if( p[i+1] > p[i]+1, c--; break)))); c}; /* _Michael Somos_, Aug 13 2017 */
%o A000009 (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^2)/eta(q))} \\ _Altug Alkan_, Mar 20 2018
%o A000009 (Magma) Coefficients(&*[1+x^m:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
%o A000009 (Haskell)
%o A000009 import Data.MemoCombinators (memo2, integral)
%o A000009 a000009 n = a000009_list !! n
%o A000009 a000009_list = map (pM 1) [0..] where
%o A000009 pM = memo2 integral integral p
%o A000009 p _ 0 = 1
%o A000009 p k m | m < k = 0
%o A000009 | otherwise = pM (k + 1) (m - k) + pM (k + 1) m
%o A000009 -- _Reinhard Zumkeller_, Sep 09 2015, Nov 05 2013
%o A000009 (Maxima) num_distinct_partitions(60,list); /* _Emanuele Munarini_, Feb 24 2014 */
%o A000009 (Maxima)
%o A000009 h(n):=if oddp(n)=true then 1 else 0;
%o A000009 S(n,m):=if n=0 then 1 else if n<m then 0 else if n=m then h(n) else sum(h(k)*S(n-k,k),k,m,n/2)+h(n);
%o A000009 makelist(S(n,1),n,0,27); /* _Vladimir Kruchinin_, Sep 07 2014 */
%o A000009 (SageMath) # uses[EulerTransform from A166861]
%o A000009 a = BinaryRecurrenceSequence(0, 1)
%o A000009 b = EulerTransform(a)
%o A000009 print([b(n) for n in range(56)]) # _Peter Luschny_, Nov 11 2020
%o A000009 (Python) # uses A010815
%o A000009 from functools import lru_cache
%o A000009 from math import isqrt
%o A000009 @lru_cache(maxsize=None)
%o A000009 def A000009(n): return 1 if n == 0 else A010815(n)+2*sum((-1)**(k+1)*A000009(n-k**2) for k in range(1,isqrt(n)+1)) # _Chai Wah Wu_, Sep 08 2021
%o A000009 (Julia) # uses A010815
%o A000009 using Memoize
%o A000009 @memoize function A000009(n)
%o A000009 n == 0 && return 1
%o A000009 s = sum((-1)^k*A000009(n - k^2) for k in 1:isqrt(n))
%o A000009 A010815(n) - 2*s
%o A000009 end # _Peter Luschny_, Sep 09 2021
%Y A000009 Apart from the first term, equals A052839-1. The rows of A053632 converge to this sequence. When reduced modulo 2 equals the absolute values of A010815. The positions of odd terms given by A001318.
%Y A000009 a(n) = Sum_{n=1..m} A097306(n, m), row sums of triangle of number of partitions of n into m odd parts.
%Y A000009 Cf. A001318, A000041, A000700, A003724, A004111, A007837, A010815, A035294, A068049, A078408, A081360, A088670, A109950, A109968, A132312, A146061, A035363, A010054, A057077, A089806, A091602, A237515, A118457 (the partitions), A118459 (partition lengths), A015723 (total number of parts), A230957 (boustrophedon transform).
%Y A000009 Cf. A167377 (complement).
%Y A000009 Cf. A067659 (odd number of parts), A067661 (even number of parts).
%Y A000009 Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.
%K A000009 nonn,core,easy,nice
%O A000009 0,4
%A A000009 _N. J. A. Sloane_
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"A306633"
] | N. J. A. Sloane | [
"easy",
"core",
"nonn",
"mult",
"nice",
"hear"
] | 0 | 5 | oeisdata/seq/A000/A000010.seq | %I A000010 M0299 N0111 #592 Mar 29 2025 18:13:56
%S A000010 1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,10,22,8,20,12,18,12,
%T A000010 28,8,30,16,20,16,24,12,36,18,24,16,40,12,42,20,24,22,46,16,42,20,32,
%U A000010 24,52,18,40,24,36,28,58,16,60,30,36,32,48,20,66,32,44
%N A000010 Euler totient function phi(n): count numbers <= n and prime to n.
%C A000010 Number of elements in a reduced residue system modulo n.
%C A000010 Degree of the n-th cyclotomic polynomial (cf. A013595). - _Benoit Cloitre_, Oct 12 2002
%C A000010 Number of distinct generators of a cyclic group of order n. Number of primitive n-th roots of unity. (A primitive n-th root x is such that x^k is not equal to 1 for k = 1, 2, ..., n - 1, but x^n = 1.) - _Lekraj Beedassy_, Mar 31 2005
%C A000010 Also number of complex Dirichlet characters modulo n; Sum_{k=1..n} a(k) is asymptotic to (3/Pi^2)*n^2. - _Steven Finch_, Feb 16 2006
%C A000010 a(n) is the highest degree of irreducible polynomial dividing 1 + x + x^2 + ... + x^(n-1) = (x^n - 1)/(x - 1). - _Alexander Adamchuk_, Sep 02 2006, corrected Sep 27 2006
%C A000010 a(p) = p - 1 for prime p. a(n) is even for n > 2. For n > 2, a(n)/2 = A023022(n) = number of partitions of n into 2 ordered relatively prime parts. - _Alexander Adamchuk_, Jan 25 2007
%C A000010 Number of automorphisms of the cyclic group of order n. - _Benoit Jubin_, Aug 09 2008
%C A000010 a(n+2) equals the number of palindromic Sturmian words of length n which are "bispecial", prefix or suffix of two Sturmian words of length n + 1. - _Fred Lunnon_, Sep 05 2010
%C A000010 Suppose that a and n are coprime positive integers, then by Euler's totient theorem, any factor of n divides a^phi(n) - 1. - _Lei Zhou_, Feb 28 2012
%C A000010 If m has k prime factors, (p_1, p_2, ..., p_k), then phi(m*n) = (Product_{i=1..k} phi (p_i*n))/phi(n)^(k-1). For example, phi(42*n) = phi(2*n)*phi(3*n)*phi(7*n)/phi(n)^2. - _Gary Detlefs_, Apr 21 2012
%C A000010 Sum_{n>=1} a(n)/n! = 1.954085357876006213144... This sum is referenced in Plouffe's inverter. - _Alexander R. Povolotsky_, Feb 02 2013 (see A336334. - _Hugo Pfoertner_, Jul 22 2020)
%C A000010 The order of the multiplicative group of units modulo n. - _Michael Somos_, Aug 27 2013
%C A000010 A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - _Michael Somos_, Dec 30 2016
%C A000010 From _Eric Desbiaux_, Jan 01 2017: (Start)
%C A000010 a(n) equals the Ramanujan sum c_n(n) (last term on n-th row of triangle A054533).
%C A000010 a(n) equals the Jordan function J_1(n) (cf. A007434, A059376, A059377, which are the Jordan functions J_2, J_3, J_4, respectively). (End)
%C A000010 For n > 1, a(n) appears to be equal to the number of semi-meander solutions for n with top arches containing exactly 2 mountain ranges and exactly 2 arches of length 1. - _Roger Ford_, Oct 11 2017
%C A000010 a(n) is the minimum dimension of a lattice able to generate, via cut-and-project, the quasilattice whose diffraction pattern features n-fold rotational symmetry. The case n=15 is the first n > 1 in which the following simpler definition fails: "a(n) is the minimum dimension of a lattice with n-fold rotational symmetry". - _Felix Flicker_, Nov 08 2017
%C A000010 Number of cyclic Latin squares of order n with the first row in ascending order. - _Eduard I. Vatutin_, Nov 01 2020
%C A000010 a(n) is the number of rational numbers p/q >= 0 (in lowest terms) such that p + q = n. - _Rémy Sigrist_, Jan 17 2021
%C A000010 From _Richard L. Ollerton_, May 08 2021: (Start)
%C A000010 Formulas for the numerous OEIS entries involving Dirichlet convolution of a(n) and some sequence h(n) can be derived using the following (n >= 1):
%C A000010 Sum_{d|n} phi(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k)) [see P. H. van der Kamp link] = Sum_{d|n} h(d)*phi(n/d) = Sum_{k=1..n} h(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). Similarly,
%C A000010 Sum_{d|n} phi(d)*h(d) = Sum_{k=1..n} h(n/gcd(n,k)) = Sum_{k=1..n} h(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)).
%C A000010 More generally,
%C A000010 Sum_{d|n} h(d) = Sum_{k=1..n} h(gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))/phi(n/gcd(n,k)).
%C A000010 In particular, for sequences involving the Möbius transform:
%C A000010 Sum_{d|n} mu(d)*h(n/d) = Sum_{k=1..n} h(gcd(n,k))*mu(n/gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} h(n/gcd(n,k))*mu(gcd(n,k))/phi(n/gcd(n,k)), where mu = A008683.
%C A000010 Use of gcd(n,k)*lcm(n,k) = n*k and phi(gcd(n,k))*phi(lcm(n,k)) = phi(n)*phi(k) provide further variations. (End)
%C A000010 From _Richard L. Ollerton_, Nov 07 2021: (Start)
%C A000010 Formulas for products corresponding to the sums above may found using the substitution h(n) = log(f(n)) where f(n) > 0 (for example, cf. formulas for the sum A018804 and product A067911 of gcd(n,k)):
%C A000010 Product_{d|n} f(n/d)^phi(d) = Product_{k=1..n} f(gcd(n,k)) = Product_{d|n} f(d)^phi(n/d) = Product_{k=1..n} f(n/gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))),
%C A000010 Product_{d|n} f(d)^phi(d) = Product_{k=1..n} f(n/gcd(n,k)) = Product_{k=1..n} f(gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))),
%C A000010 Product_{d|n} f(d) = Product_{k=1..n} f(gcd(n,k))^(1/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(1/phi(n/gcd(n,k))),
%C A000010 Product_{d|n} f(n/d)^mu(d) = Product_{k=1..n} f(gcd(n,k))^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} f(n/gcd(n,k))^(mu(gcd(n,k))/phi(n/gcd(n,k))), where mu = A008683. (End)
%C A000010 a(n+1) is the number of binary words with exactly n distinct subsequences (when n > 0). - _Radoslaw Zak_, Nov 29 2021
%D A000010 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A000010 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.
%D A000010 M. Baake and U. Grimm, Aperiodic Order Vol. 1: A Mathematical Invitation, Encyclopedia of Mathematics and its Applications 149, Cambridge University Press, 2013: see Tables 3.1 and 3.2.
%D A000010 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 193.
%D A000010 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 154-156.
%D A000010 C. W. Curtis, Pioneers of Representation Theory ..., Amer. Math. Soc., 1999; see p. 3.
%D A000010 J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Ellipses, Paris, 2004, Problème 529, pp. 71-257.
%D A000010 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, Chapter V.
%D A000010 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 115-119.
%D A000010 Carl Friedrich Gauss, "Disquisitiones Arithmeticae", Yale University Press, 1965; see p. 21.
%D A000010 Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2n-d ed.; Addison-Wesley, 1994, p. 137.
%D A000010 R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B36.
%D A000010 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 60, 62, 63, 288, 323, 328, 330.
%D A000010 Peter Hilton and Jean Pedersen, A Mathematical Tapestry, Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, pages 261-264, the Coach theorem.
%D A000010 Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.21 pp. 281-294.
%D A000010 G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, New York, Heidelberg, Berlin, 2 vols., 1976, Vol. II, problem 71, p. 126.
%D A000010 Paulo Ribenboim, The New Book of Prime Number Records.
%D A000010 Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 28-33.
%D A000010 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000010 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000010 Daniel Forgues, <a href="/A000010/b000010.txt">Table of n, phi(n) for n = 1..100000</a> (first 10000 terms from N. J. A. Sloane)
%H A000010 Milton Abramowitz and Irene A. Stegun, eds., <a href="https://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.
%H A000010 Dario A. Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method (along with sigma_0, sigma_1 and phi functions)</a>
%H A000010 Joerg Arndt, <a href="https://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 39.7, pp. 776-778.
%H A000010 F. Bayart, <a href="https://www.bibmath.net/dico/index.php?action=affiche&quoi=./i/indicateureuler.html">Indicateur d'Euler</a> (in French).
%H A000010 Alexander Bogomolny, <a href="https://www.cut-the-knot.org/blue/Euler.shtml">Euler Function and Theorem</a>.
%H A000010 Chris K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=EulersPhi">Euler's phi function</a>
%H A000010 Robert D. Carmichael, <a href="/A002180/a002180.pdf">A table of the values of m corresponding to given values of phi(m)</a>, Amer. J. Math., 30 (1908), 394-400. [Annotated scanned copy]
%H A000010 Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, <a href="https://math.dartmouth.edu/~carlp/iterate.pdf">On the normal behavior of the iterates of some arithmetic functions</a>, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
%H A000010 Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, <a href="/A000010/a000010_1.pdf">On the normal behavior of the iterates of some arithmetic functions</a>, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
%H A000010 Kevin Ford, <a href="https://arxiv.org/abs/math/9907204">The number of solutions of phi(x)=m</a>, arXiv:math/9907204 [math.NT], 1999.
%H A000010 Kevin Ford, Florian Luca and Pieter Moree, <a href="https://arxiv.org/abs/1108.3805">Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields</a>, arXiv:1108.3805 [math.NT], 2011.
%H A000010 H. Fripertinger, <a href="https://web.archive.org/web/20150910232858/http://www.uni-graz.at/~fripert/fga/k1euler.html">The Euler phi function</a>.
%H A000010 Daniele A. Gewurz and Francesca Merola, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000010 E. Pérez Herrero, <a href="https://psychedelic-geometry.blogspot.com/2010/07/totient-carnival.html">Totient Carnival partitions</a>, Psychedelic Geometry Blogspot.
%H A000010 Peter H. van der Kamp, <a href="https://arxiv.org/abs/1201.3139">On the Fourier transform of the greatest common divisor</a>, arXiv:1201.3139 [math.NT]
%H A000010 M. Lal and P. Gillard, <a href="https://dx.doi.org/10.1090/S0025-5718-69-99858-5">Table of Euler's phi function, n < 10^5</a>, Math. Comp., 23 (1969), 682-683.
%H A000010 Derrick N. Lehmer, <a href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-26/issue-3/Dicksons-History-of-the-Theory-of-Numbers/bams/1183425137.full">Review of Dickson's History of the Theory of Numbers</a>, Bull. Amer. Math. Soc., 26 (1919), 125-132.
%H A000010 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/EulerTotient">Sequences related to Euler's totient function</a>.
%H A000010 R. J. Mathar, <a href="/A000010/a000010_2.pdf">Graphical representation among sequences closely related to this one</a> (cf. N. J. A. Sloane, "Families of Essentially Identical Sequences").
%H A000010 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/301837/is-the-euler-phi-function-bounded-below">Is the Euler phi function bounded below?</a> (2013).
%H A000010 Mathforum, <a href="http://mathforum.org/library/drmath/view/51541.html">Proving phi(m) Is Even</a>.
%H A000010 Keith Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n)</a>.
%H A000010 Graeme McRae, <a href="https://web.archive.org/web/20130508193928/http://2000clicks.com/MathHelp/NumberFactorsTotientFunction.aspx">Euler's Totient Function</a>.
%H A000010 François Nicolas, <a href="https://arxiv.org/abs/0806.2068">A simple, polynomial-time algorithm for the matrix torsion problem</a>, arXiv:0806.2068 [cs.DM], 2009.
%H A000010 Matthew Parker, <a href="https://oeis.org/A000010/a000010_5M.7z">The first 5 million terms (7-Zip compressed file)</a>.
%H A000010 Carl Pomerance and Hee-Sung Yang, <a href="https://www.math.dartmouth.edu/~carlp/uupaper7.pdf">Variant of a theorem of Erdős on the sum-of-proper-divisors function</a>, Math. Comp., to appear (2014).
%H A000010 Primefan, <a href="https://primefan.tripod.com/Phi500.html">Euler's Totient Function Values For n=1 to 500, with Divisor Lists</a>.
%H A000010 Marko Riedel, <a href="https://web.archive.org/web/20170406154901/http://www.mathematik.uni-stuttgart.de/~riedelmo/combnumth.html">Combinatorics and number theory page</a>.
%H A000010 J. Barkley Rosser and Lowell Schoenfeld, <a href="https://dx.doi.org/10.1215/ijm/1255631807">Approximate formulas for some functions of prime numbers</a>, Illinois J. Math. 6 (1962), no. 1, 64-94.
%H A000010 K. Schneider, <a href="https://planetmath.org/eulerphifunction">Euler phi-function</a>, PlanetMath.org.
%H A000010 Wacław F. Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4206.pdf">Euler's Totient Function And The Theorem Of Euler</a>.
%H A000010 N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)
%H A000010 N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 14.
%H A000010 Ulrich Sondermann, <a href="https://web.archive.org/web/20110823215228/http://home.earthlink.net/~usondermann/eulertot.html">Euler's Totient Function</a>.
%H A000010 William A. Stein, <a href="https://wstein.org/edu/Fall2001/124/lectures/lecture6/html/node3.html">Phi is a Multiplicative Function</a>
%H A000010 Pinthira Tangsupphathawat, Takao Komatsu and Vichian Laohakosol, <a href="https://www.emis.de/journals/JIS/VOL21/Laohakosol/lao8.html">Minimal Polynomials of Algebraic Cosine Values, II</a>, J. Int. Seq., Vol. 21 (2018), Article 18.9.5.
%H A000010 László Tóth, <a href="http://arxiv.org/abs/1310.7053">Multiplicative arithmetic functions of several variables: a survey</a>, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
%H A000010 G. Villemin, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Nombre/TotEuler.htm">Totient d'Euler</a>.
%H A000010 K. W. Wegner, <a href="/A002180/a002180_1.pdf">Values of phi(x) = n for n from 2 through 1978</a>, mimeographed manuscript, no date. [Annotated scanned copy]
%H A000010 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ModuloMultiplicationGroup.html">Modulo Multiplication Group</a>.
%H A000010 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MoebiusTransform.html">Moebius Transform</a>.
%H A000010 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>.
%H A000010 Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler%27s_phi_function">Euler's totient function</a>.
%H A000010 Wikipedia, <a href="http://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n">Multiplicative group of integers modulo n</a>.
%H A000010 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ramanujan's_sum">Ramanujan's sum</a>
%H A000010 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/EulerPhi/03/02">First 50 values of phi(n)</a>.
%H A000010 Gang Xiao, Numerical Calculator, <a href="https://wims.univ-cotedazur.fr/wims/en_tool~number~calcnum.en.html">To display phi(n) operate on "eulerphi(n)"</a>.
%H A000010 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000010 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F A000010 phi(n) = n*Product_{distinct primes p dividing n} (1 - 1/p).
%F A000010 Sum_{d divides n} phi(d) = n.
%F A000010 phi(n) = Sum_{d divides n} mu(d)*n/d, i.e., the Moebius transform of the natural numbers; mu() = Moebius function A008683().
%F A000010 Dirichlet generating function Sum_{n>=1} phi(n)/n^s = zeta(s-1)/zeta(s). Also Sum_{n >= 1} phi(n)*x^n/(1 - x^n) = x/(1 - x)^2.
%F A000010 Multiplicative with a(p^e) = (p - 1)*p^(e-1). - _David W. Wilson_, Aug 01 2001
%F A000010 Sum_{n>=1} (phi(n)*log(1 - x^n)/n) = -x/(1 - x) for -1 < x < 1 (cf. A002088) - _Henry Bottomley_, Nov 16 2001
%F A000010 a(n) = binomial(n+1, 2) - Sum_{i=1..n-1} a(i)*floor(n/i) (see A000217 for inverse). - _Jon Perry_, Mar 02 2004
%F A000010 It is a classical result (certainly known to Landau, 1909) that lim inf n/phi(n) = 1 (taking n to be primes), lim sup n/(phi(n)*log(log(n))) = e^gamma, with gamma = Euler's constant (taking n to be products of consecutive primes starting from 2 and applying Mertens' theorem). See e.g. Ribenboim, pp. 319-320. - Pieter Moree, Sep 10 2004
%F A000010 a(n) = Sum_{i=1..n} |k(n, i)| where k(n, i) is the Kronecker symbol. Also a(n) = n - #{1 <= i <= n : k(n, i) = 0}. - _Benoit Cloitre_, Aug 06 2004 [Corrected by _Jianing Song_, Sep 25 2018]
%F A000010 Conjecture: Sum_{i>=2} (-1)^i/(i*phi(i)) exists and is approximately 0.558 (A335319). - Orges Leka (oleka(AT)students.uni-mainz.de), Dec 23 2004
%F A000010 From _Enrique Pérez Herrero_, Sep 07 2010: (Start)
%F A000010 a(n) = Sum_{i=1..n} floor(sigma_k(i*n)/sigma_k(i)*sigma_k(n)), where sigma_2 is A001157.
%F A000010 a(n) = Sum_{i=1..n} floor(tau_k(i*n)/tau_k(i)*tau_k(n)), where tau_3 is A007425.
%F A000010 a(n) = Sum_{i=1..n} floor(rad(i*n)/rad(i)*rad(n)), where rad is A007947. (End)
%F A000010 a(n) = A173557(n)*A003557(n). - _R. J. Mathar_, Mar 30 2011
%F A000010 a(n) = A096396(n) + A096397(n). - _Reinhard Zumkeller_, Mar 24 2012
%F A000010 phi(p*n) = phi(n)*(floor(((n + p - 1) mod p)/(p - 1)) + p - 1), for primes p. - _Gary Detlefs_, Apr 21 2012
%F A000010 For odd n, a(n) = 2*A135303((n-1)/2)*A003558((n-1)/2) or phi(n) = 2*c*k; the Coach theorem of Pedersen et al. Cf. A135303. - _Gary W. Adamson_, Aug 15 2012
%F A000010 G.f.: Sum_{n>=1} mu(n)*x^n/(1 - x^n)^2, where mu(n) = A008683(n). - _Mamuka Jibladze_, Apr 05 2015
%F A000010 a(n) = n - cototient(n) = n - A051953(n). - _Omar E. Pol_, May 14 2016
%F A000010 a(n) = lim_{s->1} n*zeta(s)*(Sum_{d divides n} A008683(d)/(e^(1/d))^(s-1)), for n > 1. - _Mats Granvik_, Jan 26 2017
%F A000010 Conjecture: a(n) = Sum_{a=1..n} Sum_{b=1..n} Sum_{c=1..n} 1 for n > 1. The sum is over a,b,c such that n*c - a*b = 1. - _Benedict W. J. Irwin_, Apr 03 2017
%F A000010 a(n) = Sum_{j=1..n} gcd(j, n) cos(2*Pi*j/n) = Sum_{j=1..n} gcd(j, n) exp(2*Pi*i*j/n) where i is the imaginary unit. Notice that the Ramanujan's sum c_n(k) := Sum_{j=1..n, gcd(j, n) = 1} exp(2*Pi*i*j*k/n) gives a(n) = Sum_{k|n} k*c_(n/k)(1) = Sum_{k|n} k*mu(n/k). - _Michael Somos_, May 13 2018
%F A000010 G.f.: x*d/dx(x*d/dx(log(Product_{k>=1} (1 - x^k)^(-mu(k)/k^2)))), where mu(n) = A008683(n). - _Mamuka Jibladze_, Sep 20 2018
%F A000010 a(n) = Sum_{d|n} A007431(d). - _Steven Foster Clark_, May 29 2019
%F A000010 G.f. A(x) satisfies: A(x) = x/(1 - x)^2 - Sum_{k>=2} A(x^k). - _Ilya Gutkovskiy_, Sep 06 2019
%F A000010 a(n) >= sqrt(n/2) (Nicolas). - _Hugo Pfoertner_, Jun 01 2020
%F A000010 a(n) > n/(exp(gamma)*log(log(n)) + 5/(2*log(log(n)))), except for n=223092870 (Rosser, Schoenfeld). - _Hugo Pfoertner_, Jun 02 2020
%F A000010 From _Bernard Schott_, Nov 28 2020: (Start)
%F A000010 Sum_{m=1..n} 1/a(m) = A028415(n)/A048049(n) -> oo when n->oo.
%F A000010 Sum_{n >= 1} 1/a(n)^2 = A109695.
%F A000010 Sum_{n >= 1} 1/a(n)^3 = A335818.
%F A000010 Sum_{n >= 1} 1/a(n)^k is convergent iff k > 1.
%F A000010 a(2n) = a(n) iff n is odd, and, a(2n) > a(n) iff n is even. (End) [Actually, a(2n) = 2*a(n) for even n. - _Jianing Song_, Sep 18 2022]
%F A000010 a(n) = 2*A023896(n)/n, n > 1. - _Richard R. Forberg_, Feb 03 2021
%F A000010 From _Richard L. Ollerton_, May 09 2021: (Start)
%F A000010 For n > 1, Sum_{k=1..n} phi^{(-1)}(n/gcd(n,k))*a(gcd(n,k))/a(n/gcd(n,k)) = 0, where phi^{(-1)} = A023900.
%F A000010 For n > 1, Sum_{k=1..n} a(gcd(n,k))*mu(rad(gcd(n,k)))*rad(gcd(n,k))/gcd(n,k) = 0.
%F A000010 For n > 1, Sum_{k=1..n} a(gcd(n,k))*mu(rad(n/gcd(n,k)))*rad(n/gcd(n,k))*gcd(n,k) = 0.
%F A000010 Sum_{k=1..n} a(gcd(n,k))/a(n/gcd(n,k)) = n. (End)
%F A000010 a(n) = Sum_{d|n, e|n} gcd(d, e)*mobius(n/d)*mobius(n/e) (the sum is a multiplicative function of n by Tóth, and takes the value p^e - p^(e-1) for n = p^e, a prime power). - _Peter Bala_, Jan 22 2024
%F A000010 Sum_{n >= 1} phi(n)*x^n/(1 + x^n) = x + 3*x^3 + 5*x^5 + 7*x^7 + ... = Sum_{n >= 1} phi(2*n-1)*x^(2*n-1)/(1 - x^(4*n-2)). For the first equality see Pólya and Szegő, problem 71, p. 126. - _Peter Bala_, Feb 29 2024
%F A000010 Conjecture: a(n) = lim_{k->oo} (n^(k + 1))/A000203(n^k). - _Velin Yanev_, Dec 04 2024 [A000010(p) = p-1, A000203(p^k) = (p^(k+1)-1)/(p-1), so the conjecture is true if n is prime. - _Vaclav Kotesovec_, Dec 19 2024]
%e A000010 G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 6*x^7 + 4*x^8 + 6*x^9 + 4*x^10 + ...
%e A000010 a(8) = 4 with {1, 3, 5, 7} units modulo 8. a(10) = 4 with {1, 3, 7, 9} units modulo 10. - _Michael Somos_, Aug 27 2013
%e A000010 From _Eduard I. Vatutin_, Nov 01 2020: (Start)
%e A000010 The a(5)=4 cyclic Latin squares with the first row in ascending order are:
%e A000010 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
%e A000010 1 2 3 4 0 2 3 4 0 1 3 4 0 1 2 4 0 1 2 3
%e A000010 2 3 4 0 1 4 0 1 2 3 1 2 3 4 0 3 4 0 1 2
%e A000010 3 4 0 1 2 1 2 3 4 0 4 0 1 2 3 2 3 4 0 1
%e A000010 4 0 1 2 3 3 4 0 1 2 2 3 4 0 1 1 2 3 4 0
%e A000010 (End)
%p A000010 with(numtheory): A000010 := phi; [ seq(phi(n), n=1..100) ]; # version 1
%p A000010 with(numtheory): phi := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := n*mul((1-1/t1[i][1]),i=1..nops(t1)); end; # version 2
%p A000010 # Alternative without library function:
%p A000010 A000010List := proc(N) local i, j, phi;
%p A000010 phi := Array([seq(i, i = 1 .. N+1)]);
%p A000010 for i from 2 to N + 1 do
%p A000010 if phi[i] = i then
%p A000010 for j from i by i to N + 1 do
%p A000010 phi[j] := phi[j] - iquo(phi[j], i) od
%p A000010 fi od;
%p A000010 return phi end:
%p A000010 A000010List(68); # _Peter Luschny_, Sep 03 2023
%t A000010 Array[EulerPhi, 70]
%o A000010 (Axiom) [eulerPhi(n) for n in 1..100]
%o A000010 (Magma) [ EulerPhi(n) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
%o A000010 (PARI) {a(n) = if( n==0, 0, eulerphi(n))}; /* _Michael Somos_, Feb 05 2011 */
%o A000010 (Sage) def A000010(n): return euler_phi(n) # _Jaap Spies_, Jan 07 2007
%o A000010 (Sage) [euler_phi(n) for n in range(1, 70)] # _Zerinvary Lajos_, Jun 06 2009
%o A000010 (Maxima) makelist(totient(n),n,0,1000); /* _Emanuele Munarini_, Mar 26 2011 */
%o A000010 (Haskell) a n = length (filter (==1) (map (gcd n) [1..n])) -- _Allan C. Wechsler_, Dec 29 2014
%o A000010 (Python)
%o A000010 from sympy.ntheory import totient
%o A000010 print([totient(i) for i in range(1, 70)]) # _Indranil Ghosh_, Mar 17 2017
%o A000010 (Python) # Note also the implementation in A365339.
%o A000010 (Julia) # Computes the first N terms of the sequence.
%o A000010 function A000010List(N)
%o A000010 phi = [i for i in 1:N + 1]
%o A000010 for i in 2:N + 1
%o A000010 if phi[i] == i
%o A000010 for j in i:i:N + 1
%o A000010 phi[j] -= div(phi[j], i)
%o A000010 end end end
%o A000010 return phi end
%o A000010 println(A000010List(68)) # _Peter Luschny_, Sep 03 2023
%Y A000010 Cf. A002088 (partial sums), A008683, A003434 (steps to reach 1), A007755, A049108, A002202 (values), A011755 (Sum k*phi(k)).
%Y A000010 Cf. also A005277 (nontotient numbers). For inverse see A002181, A006511, A058277.
%Y A000010 Jordan function J_k(n) is a generalization - see A059379 and A059380 (triangle of values of J_k(n)), this sequence (J_1), A007434 (J_2), A059376 (J_3), A059377 (J_4), A059378 (J_5).
%Y A000010 Cf. A054521, A023022, A054525.
%Y A000010 Row sums of triangles A134540, A127448, A143239, A143353 and A143276.
%Y A000010 Equals right and left borders of triangle A159937. - _Gary W. Adamson_, Apr 26 2009
%Y A000010 Values for prime powers p^e: A006093 (e=1), A036689 (e=2), A135177 (e=3), A138403 (e=4), A138407 (e=5), A138412 (e=6).
%Y A000010 Values for perfect powers n^e: A002618 (e=2), A053191 (e=3), A189393 (e=4), A238533 (e=5), A306411 (e=6), A239442 (e=7), A306412 (e=8), A239443 (e=9).
%Y A000010 Cf. A003558, A135303.
%Y A000010 Cf. A152455, A080737.
%Y A000010 Cf. A076479.
%Y A000010 Cf. A023900 (Dirichlet inverse of phi), A306633 (Dgf at s=3).
%K A000010 easy,core,nonn,mult,nice,hear
%O A000010 1,3
%A A000010 _N. J. A. Sloane_
| 22249ad9e32872cd726d403cb4635add |
A000011 | [
"M0312",
"N0114"
] | 103 | 2024-05-21T08:46:57 | [
"1",
"1",
"2",
"2",
"4",
"4",
"8",
"9",
"18",
"23",
"44",
"63",
"122",
"190",
"362",
"612",
"1162",
"2056",
"3914",
"7155",
"13648",
"25482",
"48734",
"92205",
"176906",
"337594",
"649532",
"1246863",
"2405236",
"4636390",
"8964800",
"17334801",
"33588234",
"65108062",
"126390032",
"245492244",
"477353376",
"928772650",
"1808676326",
"3524337980"
] | Number of n-bead necklaces (turning over is allowed) where complements are equivalent. | [
"A000011",
"A000013",
"A000117",
"A053656",
"A092668",
"A123045",
"A256216",
"A256217",
"A283846",
"A283847",
"A283848",
"A320748"
] | N. J. A. Sloane | [
"nonn",
"nice",
"easy"
] | 0 | 5 | oeisdata/seq/A000/A000011.seq | %I A000011 M0312 N0114 #103 May 21 2024 08:46:57
%S A000011 1,1,2,2,4,4,8,9,18,23,44,63,122,190,362,612,1162,2056,3914,7155,
%T A000011 13648,25482,48734,92205,176906,337594,649532,1246863,2405236,4636390,
%U A000011 8964800,17334801,33588234,65108062,126390032,245492244,477353376,928772650,1808676326,3524337980
%N A000011 Number of n-bead necklaces (turning over is allowed) where complements are equivalent.
%C A000011 a(n) is also the number of minimal fibrations of a bidirectional n-cycle over the 2-bouquet up to precompositions with automorphisms of the n-cycle and postcomposition with automorphisms of the 2-bouquet. (Boldi et al.) - _Sebastiano Vigna_, Jan 08 2018
%C A000011 For n >= 3, also the number of distinct planar embeddings of the n-sunlet graph. - _Eric W. Weisstein_, May 21 2024
%D A000011 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000011 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000011 Seiichi Manyama, <a href="/A000011/b000011.txt">Table of n, a(n) for n = 0..3335</a> (first 201 terms from T. D. Noe)
%H A000011 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>
%H A000011 Paolo Boldi and Sebastiano Vigna, <a href="https://doi.org/10.1016/S0012-365X(00)00455-6">Fibrations of Graphs</a>, Discrete Math., 243 (2002), 21-66.
%H A000011 H. Bottomley, <a href="/A000011/a000011_a000013.gif">Initial terms of A000011 and A000013</a>
%H A000011 Aharon Davidson, <a href="https://arxiv.org/abs/1907.03090">From Planck Area to Graph Theory: Topologically Distinct Black Hole Microstates</a>, arXiv:1907.03090 [gr-qc], 2019.
%H A000011 Daniel T. Eatough and Keith A. Seffen, <a href="https://doi.org/10.1115/1.4045422">Calculating the Fold Angles of Any Vertex Roof Using a Spherical Image Technique</a>, J. Mechanisms Robotics (2020) Vol. 12, No. 3, 031004.
%H A000011 N. J. Fine, <a href="http://projecteuclid.org/euclid.ijm/1255381350">Classes of periodic sequences</a>, Illinois J. Math., 2 (1958), 285-302.
%H A000011 Shinsaku Fujita, <a href="https://doi.org/10.1246/bcsj.20160369">alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method</a>, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
%H A000011 E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%H A000011 W. D. Hoskins and Anne Penfold Street, <a href="http://dx.doi.org/10.1017/S1446788700017547">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
%H A000011 W. D. Hoskins and A. P. Street, <a href="/A005513/a005513_1.pdf">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
%H A000011 Yi Hu, <a href="https://hdl.handle.net/10161/23828">Numerical Transfer Matrix Method of Next-nearest-neighbor Ising Models</a>, Master's Thesis, Duke Univ. (2021).
%H A000011 Yi Hu and Patrick Charbonneau, <a href="https://arxiv.org/abs/2106.08442">Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices</a>, arXiv:2106.08442 [cond-mat.stat-mech], 2021.
%H A000011 Karyn McLellan, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p32">Periodic coefficients and random Fibonacci sequences</a>, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
%H A000011 F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A000011 F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%H A000011 A. P. Street, <a href="/A005513/a005513.pdf">Letter to N. J. A. Sloane, N.D.</a>
%H A000011 Zhe Sun, T. Suenaga, P. Sarkar, S. Sato, M. Kotani, and H. Isobe, <a href="https://doi.org/10.1073/pnas.1606530113">Stereoisomerism, crystal structures, and dynamics of belt-shaped cyclonaphthylenes</a>, Proc. Nat. Acad. Sci. USA, vol. 113 no. 29, pp. 8109-8114, doi: 10.1073/pnas.1606530113.
%H A000011 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PlanarEmbedding.html">Planar Embedding</a>.
%H A000011 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SunletGraph.html">Sunlet Graph</a>.
%H A000011 A. Yajima, <a href="http://dx.doi.org/10.1246/bcsj.20140204">How to calculate the number of stereoisomers of inositol-homologs</a>, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264; doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text).
%H A000011 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%H A000011 <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%F A000011 a(n) = (A000013(n) + 2^floor(n/2))/2.
%e A000011 From Jason Orendorff (jason.orendorff(AT)gmail.com), Jan 09 2009: (Start)
%e A000011 The binary bracelets for small n are:
%e A000011 n: bracelets
%e A000011 0: (the empty bracelet)
%e A000011 1: 0
%e A000011 2: 00, 01
%e A000011 3: 000, 001
%e A000011 4: 0000, 0001, 0011, 0101
%e A000011 5: 00000, 00001, 00011, 00101
%e A000011 6: 000000, 000001, 000011, 000101, 000111, 001001, 001011, 010101
%e A000011 (End)
%p A000011 with(numtheory): A000011 := proc(n) local s,d; if n = 0 then RETURN(1) else s := 2^(floor(n/2)); for d in divisors(n) do s := s+(phi(2*d)*2^(n/d))/(2*n); od; RETURN(s/2); fi; end;
%t A000011 a[n_] := Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 2^Floor[n/2], Divisors[n]]/2
%t A000011 a[ n_] := If[ n < 1, Boole[n == 0], 2^Quotient[n, 2] / 2 + DivisorSum[ n, EulerPhi[2 #] 2^(n/#) &] / (4 n)]; (* _Michael Somos_, Dec 19 2014 *)
%o A000011 (PARI) {a(n) = if( n<1, n==0, 2^(n\2) / 2 + sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (4*n))}; /* _Michael Somos_, Jun 03 2002 */
%Y A000011 Column 2 of A320748.
%Y A000011 Cf. A000013. Bisections give A000117 and A092668.
%Y A000011 The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.
%K A000011 nonn,nice,easy
%O A000011 0,3
%A A000011 _N. J. A. Sloane_
%E A000011 Better description from _Christian G. Bower_
%E A000011 More terms from _David W. Wilson_, Jan 13 2000
| 7f35b4642c0810ae5c17e85923b388ba |
A000012 | [
"M0003"
] | 450 | 2025-03-15T14:36:54 | [
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
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"1",
"1",
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"1",
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"1",
"1",
"1",
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"1",
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"1",
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"1",
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"1",
"1",
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"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
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"1",
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"1",
"1",
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"1",
"1",
"1",
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"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1",
"1"
] | The simplest sequence of positive numbers: the all 1's sequence. | [
"A000004",
"A000012",
"A000027",
"A007318",
"A007395",
"A008284",
"A008287",
"A010701",
"A014410",
"A027641",
"A027907",
"A035343",
"A051801",
"A060544",
"A063260",
"A063265",
"A097805",
"A104684",
"A118800",
"A130595",
"A167374",
"A171890",
"A211216",
"A212393"
] | N. J. A. Sloane, May 16 1994 | [
"nonn",
"core",
"easy",
"mult",
"cofr",
"cons",
"tabl"
] | 0 | 5 | oeisdata/seq/A000/A000012.seq | %I A000012 M0003 #450 Mar 15 2025 14:36:54
%S A000012 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A000012 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A000012 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A000012 The simplest sequence of positive numbers: the all 1's sequence.
%C A000012 Number of ways of writing n as a product of primes.
%C A000012 Number of ways of writing n as a sum of distinct powers of 2.
%C A000012 Continued fraction for golden ratio A001622.
%C A000012 Partial sums of A000007 (characteristic function of 0). - _Jeremy Gardiner_, Sep 08 2002
%C A000012 An example of an infinite sequence of positive integers whose distinct pairwise concatenations are all primes! - _Don Reble_, Apr 17 2005
%C A000012 Binomial transform of A000007; inverse binomial transform of A000079. - _Philippe Deléham_, Jul 07 2005
%C A000012 A063524(a(n)) = 1. - _Reinhard Zumkeller_, Oct 11 2008
%C A000012 For n >= 0, let M(n) be the matrix with first row = (n n+1) and 2nd row = (n+1 n+2). Then a(n) = absolute value of det(M(n)). - _K.V.Iyer_, Apr 11 2009
%C A000012 The partial sums give the natural numbers (A000027). - _Daniel Forgues_, May 08 2009
%C A000012 From _Enrique Pérez Herrero_, Sep 04 2009: (Start)
%C A000012 a(n) is also tau_1(n) where tau_2(n) is A000005.
%C A000012 a(n) is a completely multiplicative arithmetical function.
%C A000012 a(n) is both squarefree and a perfect square. See A005117 and A000290. (End)
%C A000012 Also smallest divisor of n. - _Juri-Stepan Gerasimov_, Sep 07 2009
%C A000012 Also decimal expansion of 1/9. - _Enrique Pérez Herrero_, Sep 18 2009; corrected by _Klaus Brockhaus_, Apr 02 2010
%C A000012 a(n) is also the number of complete graphs on n nodes. - Pablo Chavez (pchavez(AT)cmu.edu), Sep 15 2009
%C A000012 Totally multiplicative sequence with a(p) = 1 for prime p. Totally multiplicative sequence with a(p) = a(p-1) for prime p. - _Jaroslav Krizek_, Oct 18 2009
%C A000012 n-th prime minus phi(prime(n)); number of divisors of n-th prime minus number of perfect partitions of n-th prime; the number of perfect partitions of n-th prime number; the number of perfect partitions of n-th noncomposite number. - _Juri-Stepan Gerasimov_, Oct 26 2009
%C A000012 For all n>0, the sequence of limit values for a(n) = n!*Sum_{k>=n} k/(k+1)!. Also, a(n) = n^0. - _Harlan J. Brothers_, Nov 01 2009
%C A000012 a(n) is also the number of 0-regular graphs on n vertices. - _Jason Kimberley_, Nov 07 2009
%C A000012 Differences between consecutive n. - _Juri-Stepan Gerasimov_, Dec 05 2009
%C A000012 From _Matthew Vandermast_, Oct 31 2010: (Start)
%C A000012 1) When sequence is read as a regular triangular array, T(n,k) is the coefficient of the k-th power in the expansion of (x^(n+1)-1)/(x-1).
%C A000012 2) Sequence can also be read as a uninomial array with rows of length 1, analogous to arrays of binomial, trinomial, etc., coefficients. In a q-nomial array, T(n,k) is the coefficient of the k-th power in the expansion of ((x^q -1)/(x-1))^n, and row n has a sum of q^n and a length of (q-1)*n + 1. (End)
%C A000012 The number of maximal self-avoiding walks from the NW to SW corners of a 2 X n grid.
%C A000012 When considered as a rectangular array, A000012 is a member of the chain of accumulation arrays that includes the multiplication table A003991 of the positive integers. The chain is ... < A185906 < A000007 < A000012 < A003991 < A098358 < A185904 < A185905 < ... (See A144112 for the definition of accumulation array.) - _Clark Kimberling_, Feb 06 2011
%C A000012 a(n) = A007310(n+1) (Modd 3) := A193680(A007310(n+1)), n>=0. For general Modd n (not to be confused with mod n) see a comment on A203571. The nonnegative members of the three residue classes Modd 3, called [0], [1], and [2], are shown in the array A088520, if there the third row is taken as class [0] after inclusion of 0. - _Wolfdieter Lang_, Feb 09 2012
%C A000012 Let M = Pascal's triangle without 1's (A014410) and V = a variant of the Bernoulli numbers A027641 but starting [1/2, 1/6, 0, -1/30, ...]. Then M*V = [1, 1, 1, 1, ...]. - _Gary W. Adamson_, Mar 05 2012
%C A000012 As a lower triangular array, T is an example of the fundamental generalized factorial matrices of A133314. Multiplying each n-th diagonal by t^n gives M(t) = I/(I-t*S) = I + t*S + (t*S)^2 + ... where S is the shift operator A129184, and T = M(1). The inverse of M(t) is obtained by multiplying the first subdiagonal of T by -t and the other subdiagonals by zero, so A167374 is the inverse of T. Multiplying by t^n/n! gives exp(t*S) with inverse exp(-t*S). - _Tom Copeland_, Nov 10 2012
%C A000012 The original definition of the meter was one ten-millionth of the distance from the Earth's equator to the North Pole. According to that historical definition, the length of one degree of latitude, that is, 60 nautical miles, would be exactly 111111.111... meters. - _Jean-François Alcover_, Jun 02 2013
%C A000012 Deficiency of 2^n. - _Omar E. Pol_, Jan 30 2014
%C A000012 Consider n >= 1 nonintersecting spheres each with surface area S. Define point p on sphere S_i to be a "public point" if and only if there exists a point q on sphere S_j, j != i, such that line segment pq INTERSECT S_i = {p} and pq INTERSECT S_j = {q}; otherwise, p is a "private point". The total surface area composed of exactly all private points on all n spheres is a(n)*S = S. ("The Private Planets Problem" in Zeitz.) - _Rick L. Shepherd_, May 29 2014
%C A000012 For n>0, digital roots of centered 9-gonal numbers (A060544). - _Colin Barker_, Jan 30 2015
%C A000012 Product of nonzero digits in base-2 representation of n. - _Franklin T. Adams-Watters_, May 16 2016
%C A000012 Alternating row sums of triangle A104684. - _Wolfdieter Lang_, Sep 11 2016
%C A000012 A fixed point of the run length transform. - _Chai Wah Wu_, Oct 21 2016
%C A000012 Length of period of continued fraction for sqrt(A002522) or sqrt(A002496). - _A.H.M. Smeets_, Oct 10 2017
%C A000012 a(n) is also the determinant of the (n+1) X (n+1) matrix M defined by M(i,j) = binomial(i,j) for 0 <= i,j <= n, since M is a lower triangular matrix with main diagonal all 1's. - _Jianing Song_, Jul 17 2018
%C A000012 a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = min(i,j) for 1 <= i,j <= n (see Xavier Merlin reference). - _Bernard Schott_, Dec 05 2018
%C A000012 a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = tau(gcd(i,j)) for 1 <= i,j <= n (see De Koninck & Mercier reference). - _Bernard Schott_, Dec 08 2020
%D A000012 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
%D A000012 J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 692 pp. 90 and 297, Ellipses, Paris, 2004.
%D A000012 Xavier Merlin, Méthodix Algèbre, Exercice 1-a), page 153, Ellipses, Paris, 1995.
%D A000012 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000012 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%D A000012 Paul Zeitz, The Art and Craft of Mathematical Problem Solving, The Great Courses, The Teaching Company, 2010 (DVDs and Course Guidebook, Lecture 6: "Pictures, Recasting, and Points of View", pp. 32-34).
%H A000012 Charles R Greathouse IV, <a href="/A000012/b000012.txt">Table of n, a(n) for n = 0..10000</a> [Useful when <a href="/plot2.html">plotting one sequence against another</a>.]
%H A000012 Jeremiah Bartz, Bruce Dearden, and Joel Iiams, <a href="https://arxiv.org/abs/1810.07895">Classes of Gap Balancing Numbers</a>, arXiv:1810.07895 [math.NT], 2018.
%H A000012 Harlan Brothers, <a href="http://functions.wolfram.com/GammaBetaErf/Factorial/23/01/0002/">Factorial: Summation (formula 06.01.23.0002)</a>, The Wolfram Functions Site - _Harlan J. Brothers_, Nov 01 2009
%H A000012 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seqs., Vol. 6, 2003.
%H A000012 A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, <a href="http://dx.doi.org/10.1007/978-3-0348-0237-6">The Tower of Hanoi - Myths and Maths</a>, Birkhäuser 2013. See page 172. <a href="http://tohbook.info">Book's website</a>
%H A000012 L. B. W. Jolley, <a href="https://archive.org/details/summationofserie00joll">Summation of Series</a>, Dover, 1961
%H A000012 Jerry Metzger and Thomas Richards, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Metzger/metz1.html">A Prisoner Problem Variation</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
%H A000012 László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Nemeth/nemeth6.html">The trinomial transform triangle</a>, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also <a href="https://arxiv.org/abs/1807.07109">arXiv:1807.07109</a> [math.NT], 2018.
%H A000012 Robert Price, <a href="/A000012/a000012_1.txt">Comments on A000012 concerning Elementary Cellular Automata</a>, Jan 31 2016
%H A000012 N. J. A. Sloane, <a href="/A000012/a000012.html">Illustration of initial terms</a>
%H A000012 Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%H A000012 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>
%H A000012 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticNumber.html">Chromatic Number</a>
%H A000012 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H A000012 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A000012 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A000012 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A000012 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000012 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H A000012 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>
%H A000012 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A000012 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H A000012 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%H A000012 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%F A000012 a(n) = 1.
%F A000012 G.f.: 1/(1-x).
%F A000012 E.g.f.: exp(x).
%F A000012 G.f.: Product_{k>=0} (1 + x^(2^k)). - _Zak Seidov_, Apr 06 2007
%F A000012 Completely multiplicative with a(p^e) = 1.
%F A000012 Regarded as a square array by antidiagonals, g.f. 1/((1-x)(1-y)), e.g.f. Sum T(n,m) x^n/n! y^m/m! = e^{x+y}, e.g.f. Sum T(n,m) x^n y^m/m! = e^y/(1-x). Regarded as a triangular array, g.f. 1/((1-x)(1-xy)), e.g.f. Sum T(n,m) x^n y^m/m! = e^{xy}/(1-x). - _Franklin T. Adams-Watters_, Feb 06 2006
%F A000012 Dirichlet g.f.: zeta(s). - _Ilya Gutkovskiy_, Aug 31 2016
%F A000012 a(n) = Sum_{l=1..n} (-1)^(l+1)*2*cos(Pi*l/(2*n+1)) = 1 identically in n >= 1 (for n=0 one has 0 from the undefined sum). From the Jolley reference, (429) p. 80. Interpretation: consider the n segments between x=0 and the n positive zeros of the Chebyshev polynomials S(2*n, x) (see A049310). Then the sum of the lengths of every other segment starting with the one ending in the largest zero (going from the right to the left) is 1. - _Wolfdieter Lang_, Sep 01 2016
%F A000012 As a lower triangular matrix, T = M*T^(-1)*M = M*A167374*M, where M(n,k) = (-1)^n A130595(n,k). Note that M = M^(-1). Cf. A118800 and A097805. - _Tom Copeland_, Nov 15 2016
%e A000012 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...)))) = A001622.
%e A000012 1/9 = 0.11111111111111...
%e A000012 From _Wolfdieter Lang_, Feb 09 2012: (Start)
%e A000012 Modd 7 for nonnegative odd numbers not divisible by 3:
%e A000012 A007310: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, ...
%e A000012 Modd 3: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A000012 (End)
%p A000012 seq(1, i=0..150);
%t A000012 Array[1 &, 50] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
%o A000012 (Magma) [1 : n in [0..100]];
%o A000012 (PARI) {a(n) = 1};
%o A000012 (Haskell)
%o A000012 a000012 = const 1
%o A000012 a000012_list = repeat 1 -- _Reinhard Zumkeller_, May 07 2012
%o A000012 (Maxima) makelist(1, n, 1, 30); /* _Martin Ettl_, Nov 07 2012 */
%o A000012 (Python) print([1 for n in range(90)]) # _Michael S. Branicky_, Apr 04 2022
%Y A000012 Cf. A000004, A007395, A010701, A000027, A027641, A014410, A211216, A212393, A060544, A051801, A104684.
%Y A000012 For other q-nomial arrays, see A007318, A027907, A008287, A035343, A063260, A063265, A171890. - _Matthew Vandermast_, Oct 31 2010
%Y A000012 Cf. A097805, A118800, A130595, A167374, A008284 (multisets).
%K A000012 nonn,core,easy,mult,cofr,cons,tabl
%O A000012 0,1
%A A000012 _N. J. A. Sloane_, May 16 1994
| 6b30fc6b997ccaa842397602d6270606 |
A000013 | [
"M0313",
"N0115"
] | 103 | 2024-11-26T13:40:50 | [
"1",
"1",
"2",
"2",
"4",
"4",
"8",
"10",
"20",
"30",
"56",
"94",
"180",
"316",
"596",
"1096",
"2068",
"3856",
"7316",
"13798",
"26272",
"49940",
"95420",
"182362",
"349716",
"671092",
"1290872",
"2485534",
"4794088",
"9256396",
"17896832",
"34636834",
"67110932",
"130150588",
"252648992",
"490853416",
"954444608",
"1857283156",
"3616828364"
] | Definition (1): Number of n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed. | [
"A000010",
"A000013",
"A000016",
"A000031",
"A000116",
"A027750",
"A128976"
] | N. J. A. Sloane | [
"nonn",
"nice",
"easy"
] | 0 | 5 | oeisdata/seq/A000/A000013.seq | %I A000013 M0313 N0115 #103 Nov 26 2024 13:40:50
%S A000013 1,1,2,2,4,4,8,10,20,30,56,94,180,316,596,1096,2068,3856,7316,13798,
%T A000013 26272,49940,95420,182362,349716,671092,1290872,2485534,4794088,
%U A000013 9256396,17896832,34636834,67110932,130150588,252648992,490853416,954444608,1857283156,3616828364
%N A000013 Definition (1): Number of n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed.
%C A000013 Definition (2): Equivalently, number of different output sequences from an n-stage pure cycling shift register when 2 sequences are considered the same if one is the complement of the other.
%C A000013 Definition (3): Also number of different output sequences from an n-stage pure cycling shift register constrained so contents have even weight.
%C A000013 Definition (4): Also number of output sequences from (n-1)-stage shift register which feeds back the mod 2 sum of the contents of the register.
%C A000013 The equivalence of definitions (1) and (2) follows at once from the definitions.
%C A000013 If u is an output sequence of type (2) then its derivative is of type (3) - so (2) and (3) count the same things.
%C A000013 If we have a shift register of type (4), append a new cell which contains the mod 2 sum of the contents to get a shift register of type (3). So (3) and (4) count the same things.
%C A000013 If n is even, a(n) = A000116(n/2). If 2^(n+1)-1 is prime, then a(n) = A128976(n+1), the number of cycles in the digraph of the Lucas-Lehmer operator LL(x) = x^2 - 2 acting on Z/(2^(n+1)-1). - _M. F. Hasler_, May 19 2007
%C A000013 Also number of 2n-bead balanced binary necklaces that are equivalent to their complements. - _Andrew Howroyd_, Sep 29 2017
%D A000013 S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.
%D A000013 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000013 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000013 Seiichi Manyama, <a href="/A000013/b000013.txt">Table of n, a(n) for n = 0..3334</a> (first 201 terms from T. D. Noe)
%H A000013 Nicolás Álvarez, Victória Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, <a href="https://www-2.dc.uba.ar/staff/becher/papers/extremal.pdf">On extremal factors of de Bruijn-like graphs</a>, Univ. Buenos Aires (Argentina 2023). See also <a href="https://arxiv.org/abs/2308.16257">arXiv:2308.16257</a> [math.CO], 2023. See references.
%H A000013 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p. 151, p. 408.
%H A000013 Henry Bottomley, <a href="/A000011/a000011_a000013.gif">Initial terms of A000011 and A000013</a>
%H A000013 Zachary E. Chin and Isaac L. Chuang, <a href="https://arxiv.org/abs/2410.00893">The quantum trajectory sensing problem and its solution</a>, arXiv:2410.00893 [quant-ph], 2024. See p. 19.
%H A000013 N. J. Fine, <a href="http://projecteuclid.org/euclid.ijm/1255381350">Classes of periodic sequences</a>, Illinois J. Math., 2 (1958), 285-302.
%H A000013 E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%H A000013 Darij Grinberg and Peter Mao, <a href="https://arxiv.org/abs/2405.08937">Necklaces over a group with identity product</a>, arXiv:2405.08937 [math.CO], 2024. See pp. 15, 22.
%H A000013 Karyn McLellan, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p32">Periodic coefficients and random Fibonacci sequences</a>, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
%H A000013 Frank Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A000013 Frank Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%H A000013 N. J. A. Sloane, <a href="http://neilsloane.com/doc/dijen.txt">On single-deletion-correcting codes</a>
%H A000013 N. J. A. Sloane, <a href="http://arxiv.org/abs/math/0207197">On single-deletion-correcting codes</a>, arXiv:math/0207197 [math.CO], 2002; in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
%H A000013 N. J. A. Sloane, <a href="/A000013/a000013.txt">Maple code for this and related sequences</a>
%H A000013 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%F A000013 a(n) = Sum_{ d divides n } (phi(2*d)*2^(n/d))/(2*n) for n>0. - _Michael Somos_, Oct 20 1999
%F A000013 G.f.: 1 - Sum_{i>=1} phi(2*i)*log(1-2*x^i)/(2*i). - _Herbert Kociemba_, Nov 01 2016
%F A000013 From _Richard L. Ollerton_, May 11 2021: (Start)
%F A000013 For n >= 1:
%F A000013 a(n) = (1/(2*n))*Sum_{k=1..n} phi(2*gcd(n,k))*2^(n/gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010.
%F A000013 a(n) = (1/(2*n))*Sum_{k=1..n} phi(2*n/gcd(n,k))*2^gcd(n,k)/phi(n/gcd(n,k)). (End)
%F A000013 a(n) ~ 2^(n-1)/n. - _Cedric Lorand_, Apr 24 2022
%e A000013 G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 10*x^7 + 20*x^8 + ...
%p A000013 with(numtheory): A000013 := proc(n) local s,d; if n = 0 then RETURN(1) else s := 0; for d in divisors(n) do s := s+(phi(2*d)*2^(n/d))/(2*n); od; RETURN(s); fi; end;
%t A000013 a[n_] := Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 0, Divisors[n]]
%t A000013 a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, EulerPhi[2 #] 2^(n/#) &] / (2 n)]; (* _Michael Somos_, Dec 19 2014 *)
%t A000013 mx=40;CoefficientList[Series[1-Sum[EulerPhi[2i] Log[1-2*x^i]/(2i),{i,1,mx}],{x,0,mx}],x] (* _Herbert Kociemba_, Nov 01 2016 *)
%o A000013 (PARI) {a(n) = if( n<1, n==0, sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (2*n))}; /* _Michael Somos_, Oct 20 1999 */
%o A000013 (Haskell)
%o A000013 a000013 0 = 1
%o A000013 a000013 n = sum (zipWith (*)
%o A000013 (map (a000010 . (* 2)) ds) (map (2 ^) $ reverse ds)) `div` (2 * n)
%o A000013 where ds = a027750_row n
%o A000013 -- _Reinhard Zumkeller_, Jul 08 2013
%o A000013 (Python)
%o A000013 from sympy import divisors, totient
%o A000013 def a(n): return 1 if n<1 else sum([totient(2*d)*2**(n/d) for d in divisors(n)])/(2*n) # _Indranil Ghosh_, Apr 28 2017
%Y A000013 Cf. A000031, A000016, A000116.
%Y A000013 Cf. A128976.
%Y A000013 Cf. A000010, A027750.
%K A000013 nonn,nice,easy
%O A000013 0,3
%A A000013 _N. J. A. Sloane_
| 0b6fcd028c981d3d910e5d1da5f44419 |
A000014 | [
"M0320",
"N0118"
] | 100 | 2025-02-16T08:32:18 | [
"0",
"1",
"1",
"0",
"1",
"1",
"2",
"2",
"4",
"5",
"10",
"14",
"26",
"42",
"78",
"132",
"249",
"445",
"842",
"1561",
"2988",
"5671",
"10981",
"21209",
"41472",
"81181",
"160176",
"316749",
"629933",
"1256070",
"2515169",
"5049816",
"10172638",
"20543579",
"41602425",
"84440886",
"171794492",
"350238175",
"715497037",
"1464407113"
] | Number of series-reduced trees with n nodes. | [
"A000014",
"A000055",
"A001678",
"A007827",
"A271205"
] | N. J. A. Sloane | [
"nonn",
"easy",
"core",
"nice"
] | 0 | 5 | oeisdata/seq/A000/A000014.seq | %I A000014 M0320 N0118 #100 Feb 16 2025 08:32:18
%S A000014 0,1,1,0,1,1,2,2,4,5,10,14,26,42,78,132,249,445,842,1561,2988,5671,
%T A000014 10981,21209,41472,81181,160176,316749,629933,1256070,2515169,5049816,
%U A000014 10172638,20543579,41602425,84440886,171794492,350238175,715497037,1464407113
%N A000014 Number of series-reduced trees with n nodes.
%C A000014 Other terms for "series-reduced tree": (i) homeomorphically irreducible tree, (ii) homeomorphically reduced tree, (iii) reduced tree, (iv) topological tree.
%C A000014 In a series-reduced tree, vertices cannot have degree 2; they can be leaves or have >= 2 branches.
%D A000014 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 284.
%D A000014 D. G. Cantor, personal communication.
%D A000014 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
%D A000014 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62, Fig. 3.3.3.
%D A000014 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 526.
%D A000014 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000014 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000014 Matthew Parker, <a href="/A000014/b000014.txt">Table of n, a(n) for n = 0..1000</a> (first 501 terms from Christian G. Bower)
%H A000014 David Callan, <a href="http://arxiv.org/abs/1406.7784">A sign-reversing involution to count labeled lone-child-avoiding trees</a>, arXiv:1406.7784 [math.CO], 30 June 2014.
%H A000014 Ira M. Gessel, <a href="https://arxiv.org/abs/2305.03157">Good Will Hunting's Problem: Counting Homeomorphically Irreducible Trees</a>, arXiv:2305.03157 [math.CO], 2023.
%H A000014 James Grime and Brady Haran, <a href="http://www.youtube.com/watch?v=iW_LkYiuTKE">The problem in Good Will Hunting</a>, 2013 (Numberphile video).
%H A000014 Frank Harary and Geert Prins, <a href="http://dx.doi.org/10.1007/BF02559543">The number of homeomorphically irreducible trees and other species</a>, Acta Math., 101 (1959), 141-162.
%H A000014 F. Harary, R. W. Robinson and A. J. Schwenk, <a href="http://dx.doi.org/10.1017/S1446788700016190">Twenty-step algorithm for determining the asymptotic number of trees of various species</a>, J. Austral. Math. Soc., Series A, 20 (1975), 483-503.
%H A000014 F. Harary, R. W. Robinson and A. J. Schwenk, <a href="http://dx.doi.org/10.1017/S1446788700033760">Corrigenda: Twenty-step algorithm for determining the asymptotic number of trees of various species</a>, J. Austral. Math. Soc., Series A 41 (1986), p. 325.
%H A000014 P. Leroux and B. Miloudi, <a href="/A000081/a000081_2.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
%H A000014 B. D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/data/trees.html">Lists of Trees sorted by diameter and Homeomorphically irreducible trees, with <= 22 nodes.</a>
%H A000014 B. D. McKay, <a href="/A000014/a000014.pdf">Lists of Trees sorted by diameter and Homeomorphically irreducible trees, with <= 22 nodes.</a> [Cached copy of top page only, pdf file, no active links, with permission]
%H A000014 Matthew Parker, <a href="https://oeis.org/A000014/a000014_2K.7z">The first 2000 terms (7-Zip compressed file)</a>
%H A000014 A. J. Schwenk, <a href="/A002988/a002988.pdf">Letter to N. J. A. Sloane, Aug 1972</a>
%H A000014 N. J. A. Sloane, <a href="/A000014/a000014.gif">Illustration of initial terms</a>
%H A000014 Peter Steinbach, <a href="/A000088/a000088_17.pdf">Field Guide to Simple Graphs, Volume 1</a>, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
%H A000014 Peter Steinbach, <a href="/A000055/a000055_12.pdf">Field Guide to Simple Graphs, Volume 3</a>, Part 12 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
%H A000014 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Series-ReducedTree.html">Series-Reduced Tree</a>
%H A000014 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%H A000014 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F A000014 G.f.: A(x) = ((x-1)/x)*f(x) + ((1+x)/x^2)*g(x) - (1/x^2)*g(x)^2 where f(x) is g.f. for A059123 and g(x) is g.f. for A001678. [Harary and E. M. Palmer, p. 62, Eq. (3.3.10) with extra -(1/x^2)*Hbar(x)^2 term which should be there according to eq.(3.3.14), p. 63, with eq.(3.3.9)]. [corrected by _Wolfdieter Lang_, Jan 09 2001]
%F A000014 a(n) ~ c * d^n / n^(5/2), where d = A246403 = 2.189461985660850..., c = 0.684447272004914061023163279794145361469033868145768075109924585532604582794... - _Vaclav Kotesovec_, Aug 25 2014
%e A000014 G.f. = x + x^2 + x^4 + x^5 + 2*x^6 + 2*x^7 + 4*x^8 + 5*x^9 + 10*x^10 + ...
%e A000014 The star graph with n nodes (except for n=3) is a series-reduced tree. For n=6 the other series-reduced tree is shaped like the letter H. - _Michael Somos_, Dec 19 2014
%p A000014 with(powseries): with(combstruct): n := 30: Order := n+3: sys := {B = Prod(C,Z), S = Set(B,1 <= card), C = Union(Z,S)}:
%p A000014 G001678 := (convert(gfseries(sys,unlabeled,x) [S(x)], polynom)) * x^2: G0temp := G001678 + x^2:
%p A000014 G059123 := G0temp / x + G0temp - (G0temp^2+eval(G0temp,x=x^2))/(2*x):
%p A000014 G000014 := ((x-1)/x) * G059123 + ((1+x)/x^2) * G0temp - (1/x^2) * G0temp^2:
%p A000014 A000014 := 0,seq(coeff(G000014,x^i),i=1..n); # Ulrich Schimke (ulrschimke(AT)aol.com)
%t A000014 a[n_] := If[n<1, 0, A = x/(1-x^2) + x*O[x]^n; For[k=3, k <= n-1, k++, A = A/(1 - x^k + x*O[x]^n)^SeriesCoefficient[A, k]]; s = ((Normal[A] /. x -> x^2) + O[x]^(2n))*(1-x) + A*(2-A)*(1+x); SeriesCoefficient[s, n]/2]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 02 2016, adapted from PARI *)
%o A000014 (PARI) {a(n) = my(A); if( n<1, 0, A = x / (1 - x^2) + x * O(x^n); for(k=3, n-1, A /= (1 - x^k + x * O(x^n))^polcoeff(A, k)); polcoeff( (subst(A, x, x^2) * (1 - x) + A * (2 - A) * (1 + x)) / 2, n))}; /* _Michael Somos_, Dec 19 2014 */
%Y A000014 Cf. A000055 (trees), A001678 (series-reduced planted trees), A007827 (series-reduced trees by leaves), A271205 (series-reduced trees by leaves and nodes).
%K A000014 nonn,easy,core,nice
%O A000014 0,7
%A A000014 _N. J. A. Sloane_
| 206781371efee6d268af8cc32d42bb50 |
A000015 | null | 78 | 2025-02-16T08:32:18 | [
"1",
"2",
"3",
"4",
"5",
"7",
"7",
"8",
"9",
"11",
"11",
"13",
"13",
"16",
"16",
"16",
"17",
"19",
"19",
"23",
"23",
"23",
"23",
"25",
"25",
"27",
"27",
"29",
"29",
"31",
"31",
"32",
"37",
"37",
"37",
"37",
"37",
"41",
"41",
"41",
"41",
"43",
"43",
"47",
"47",
"47",
"47",
"49",
"49",
"53",
"53",
"53",
"53",
"59",
"59",
"59",
"59",
"59",
"59",
"61",
"61",
"64",
"64",
"64",
"67",
"67",
"67",
"71",
"71",
"71",
"71",
"73"
] | Smallest prime power >= n. | [
"A000015",
"A000961",
"A031218"
] | N. J. A. Sloane | [
"nonn",
"easy"
] | 0 | 5 | oeisdata/seq/A000/A000015.seq | %I A000015 #78 Feb 16 2025 08:32:18
%S A000015 1,2,3,4,5,7,7,8,9,11,11,13,13,16,16,16,17,19,19,23,23,23,23,25,25,27,
%T A000015 27,29,29,31,31,32,37,37,37,37,37,41,41,41,41,43,43,47,47,47,47,49,49,
%U A000015 53,53,53,53,59,59,59,59,59,59,61,61,64,64,64,67,67,67,71,71,71,71,73
%N A000015 Smallest prime power >= n.
%C A000015 The length of the m-th run of {a(n)} is the length of the (m-1)-st run of A031218 for m > 1. - _Colin Linzer_, Mar 08 2024
%H A000015 David W. Wilson, <a href="/A000015/b000015.txt">Table of n, a(n) for n = 1..10000</a>
%H A000015 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">Prime Power</a>.
%F A000015 a(A110654(n+1)) = A188666(n). - _Reinhard Zumkeller_, Apr 25 2011, corrected by _M. F. Hasler_, Jul 25 2015
%F A000015 a(n) = A188666(2n-1). - _M. F. Hasler_, Jul 25 2015
%p A000015 N:= 1000: # to get all terms <= N
%p A000015 Primes:= select(isprime,{$1..N}):
%p A000015 PPs:= {1} union Primes:
%p A000015 for k from 1 to ilog2(N) do
%p A000015 PPs:= PPs union map(`^`, select(`<=`,Primes, floor(N^(1/k))),k)
%p A000015 od:
%p A000015 PPs:= sort(convert(PPs,list)):
%p A000015 1, seq(PPs[i]$(PPs[i]-PPs[i-1]), i=2..nops(PPs)); # _Robert Israel_, Jul 23 2015
%t A000015 Insert[Table[m:=n;While[Not[Length[FactorInteger[m]]==1],m++ ];m,{n,2,100}], 1, 1] (* _Stefan Steinerberger_, Apr 17 2006 *)
%t A000015 a[n_] := NestWhile[# + 1 &, n, Not@*PrimePowerQ]; (* _Matthew House_, Jul 14 2015, v6.0+ *)
%t A000015 a[ n_] := If[ n < 2, Boole[n == 1], Module[{m = n}, While[ ! PrimePowerQ[ m], m++]; m]]; (* _Michael Somos_, Mar 06 2018 *)
%t A000015 a[ n_] := If[ n < 1, 0, Module[{m = n}, While[ Length[ FactorInteger @ m ] != 1, m++]; m]]; (* _Michael Somos_, Mar 06 2018 *)
%o A000015 (PARI) {a(n) = if( n<1, 0, while(matsize(factor(n))[1]>1, n++); n)}; /* _Michael Somos_, Jul 16 2002 */
%o A000015 (PARI) a(n)=if(n>1,while(!isprimepower(n),n++));n \\ _Charles R Greathouse IV_, Feb 01 2013
%o A000015 (Sage) [next_prime_power(n) for n in range(72)] # _Zerinvary Lajos_, Jun 13 2009
%o A000015 (Haskell)
%o A000015 a000015 n = a000015_list !! (n-1)
%o A000015 a000015_list = 1 : concat
%o A000015 (zipWith(\pp qq -> replicate (fromInteger (pp - qq)) pp)
%o A000015 (tail a000961_list) a000961_list)
%o A000015 -- _Reinhard Zumkeller_, Nov 17 2011, Apr 25 2011
%o A000015 (Python)
%o A000015 from itertools import count
%o A000015 from sympy import factorint
%o A000015 def A000015(n): return next(filter(lambda m:len(factorint(m))<=1, count(n))) # _Chai Wah Wu_, Oct 25 2024
%Y A000015 Cf. A000961, A031218.
%K A000015 nonn,easy
%O A000015 1,2
%A A000015 _N. J. A. Sloane_
%E A000015 More terms from _Michael Somos_, Jul 16 2002
| 10d73cc87131d78cad038b7542cf36aa |
A000016 | [
"M0324",
"N0121"
] | 143 | 2025-04-16T03:02:47 | [
"1",
"1",
"1",
"2",
"2",
"4",
"6",
"10",
"16",
"30",
"52",
"94",
"172",
"316",
"586",
"1096",
"2048",
"3856",
"7286",
"13798",
"26216",
"49940",
"95326",
"182362",
"349536",
"671092",
"1290556",
"2485534",
"4793492",
"9256396",
"17895736",
"34636834",
"67108864",
"130150588",
"252645136",
"490853416"
] | a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage. | [
"A000013",
"A000016",
"A000031",
"A000048",
"A051293",
"A053633",
"A053634",
"A065795",
"A068009",
"A082550",
"A135342",
"A182469",
"A237984",
"A240850",
"A327471",
"A327477",
"A327478"
] | N. J. A. Sloane | [
"nonn",
"nice",
"easy",
"changed"
] | 0 | 5 | oeisdata/seq/A000/A000016.seq | %I A000016 M0324 N0121 #143 Apr 16 2025 03:02:47
%S A000016 1,1,1,2,2,4,6,10,16,30,52,94,172,316,586,1096,2048,3856,7286,13798,
%T A000016 26216,49940,95326,182362,349536,671092,1290556,2485534,4793492,
%U A000016 9256396,17895736,34636834,67108864,130150588,252645136,490853416
%N A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.
%C A000016 Also a(n+1) = number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the sum of its contents. E.g., for n=5 there are 6 such sequences.
%C A000016 Also a(n+1) = number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 0 (mod n+1) = size of Varshamov-Tenengolts code VT_0(n). E.g., |VT_0(5)| = 6 = a(6).
%C A000016 Number of binary necklaces with an odd number of zeros. - _Joerg Arndt_, Oct 26 2015
%C A000016 Also, number of subsets of {1,2,...,n-1} which sum to 0 modulo n (cf. A063776). - _Max Alekseyev_, Mar 26 2016
%C A000016 From _Gus Wiseman_, Sep 14 2019: (Start)
%C A000016 Also the number of subsets of {1..n} containing n whose mean is an element. For example, the a(1) = 1 through a(8) = 16 subsets are:
%C A000016 1 2 3 4 5 6 7 8
%C A000016 123 234 135 246 147 258
%C A000016 345 456 357 468
%C A000016 12345 1236 567 678
%C A000016 1456 2347 1348
%C A000016 23456 2567 1568
%C A000016 12467 3458
%C A000016 13457 3678
%C A000016 34567 12458
%C A000016 1234567 14578
%C A000016 23578
%C A000016 24568
%C A000016 45678
%C A000016 123468
%C A000016 135678
%C A000016 2345678
%C A000016 (End)
%C A000016 Number of self-dual binary necklaces with 2n beads (cf. A263768, A007147). - _Bernd Mulansky_, Apr 25 2023
%D A000016 B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
%D A000016 S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.
%D A000016 J. Hedetniemi and K. R. Hutson, Equilibrium of shortest path load in ring network, Congressus Numerant., 203 (2010), 75-95. See p. 83.
%D A000016 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000016 N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
%D A000016 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000016 D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Diff. Eqs. 20 (1) (2008) 201, eq. (39)
%H A000016 Seiichi Manyama, <a href="/A000016/b000016.txt">Table of n, a(n) for n = 0..3334</a> (first 201 terms from T. D. Noe)
%H A000016 Nicolás Álvarez, Victória Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, <a href="https://www-2.dc.uba.ar/staff/becher/papers/extremal.pdf">On extremal factors of de Bruijn-like graphs</a>, Univ. Buenos Aires (Argentina 2023). See also <a href="https://arxiv.org/abs/2308.16257">arXiv:2308.16257</a> [math.CO], 2023. See references.
%H A000016 Joshua P. Bowman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Bowman/bowman4.html">Compositions with an Odd Number of Parts, and Other Congruences</a>, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 17.
%H A000016 A. E. Brouwer, <a href="https://ir.cwi.nl/pub/6805">The Enumeration of Locally Transitive Tournaments</a>, Math. Centr. Report ZW138, Amsterdam, 1980.
%H A000016 S. Butenko, P. Pardalos, I. Sergienko, V. P. Shylo and P. Stetsyuk, <a href="http://dx.doi.org/10.1007/978-0-387-98096-6_12">Estimating the size of correcting codes using extremal graph problems</a>, Optimization, 227-243, Springer Optim. Appl., 32, Springer, New York, 2009.
%H A000016 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000016 Sébastien Designolle, Tamás Vértesi, and Sebastian Pokutta, <a href="https://arxiv.org/abs/2310.20677">Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms</a>, arXiv:2310.20677 [quant-ph], 2023.
%H A000016 T. M. A. Fink, <a href="https://arxiv.org/abs/2302.05314">Exact dynamics of the critical Kauffman model with connectivity one</a>, arXiv:2302.05314 [cond-mat.stat-mech], 2023.
%H A000016 R. W. Hall and P. Klingsberg, <a href="http://www.jstor.org/stable/27642087">Asymmetric rhythms and tiling canons</a>, Amer. Math. Monthly, 113 (2006), 887-896.
%H A000016 A. A. Kulkarni, N. Kiyavash and R. Sreenivas, <a href="http://www.sc.iitb.ac.in/~ankur/docs/CombinInsightDM_final.pdf">On the Varshamov-Tenengolts Construction on Binary Strings</a>, 2013.
%H A000016 E. M. Palmer and R. W. Robinson, <a href="http://projecteuclid.org/euclid.pjm/1102711113">Enumeration of self-dual configurations</a>, Pacific J. Math., 110 (1984), 203-221.
%H A000016 R. Pries and C. Weir, <a href="http://arxiv.org/abs/1302.6261">The Ekedahl-Oort type of Jacobians of Hermitian curves</a>, arXiv preprint arXiv:1302.6261 [math.NT], 2013.
%H A000016 N. J. A. Sloane, <a href="http://neilsloane.com/doc/dijen.txt">On single-deletion-correcting codes</a>
%H A000016 N. J. A. Sloane, <a href="/A265032/a265032.html">Challenge Problems: Independent Sets in Graphs</a>
%H A000016 Yan Bo Ti, Gabriel Verret, and Lukas Zobernig, <a href="https://arxiv.org/abs/2203.08401">Abelian Varieties with p-rank Zero</a>, arXiv:2203.08401 [math.NT], 2022.
%H A000016 Antonio Vera López, Luis Martínez, Antonio Vera Pérez, Beatriz Vera Pérez, and Olga Basova, <a href="https://doi.org/10.1016/j.laa.2017.05.027">Combinatorics related to Higman's conjecture I: Parallelogramic digraphs and dispositions</a>, Linear Algebra Appl. 530, 414-444 (2017).
%H A000016 <a href="/index/To#tournament">Index entries for sequences related to tournaments</a>
%H A000016 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%H A000016 <a href="/index/Su#subsetsums">Index entries for sequences related to subset sums modulo m</a>
%F A000016 a(n) = Sum_{odd d divides n} (phi(d)*2^(n/d))/(2*n), n>0.
%F A000016 a(n) = A063776(n)/2.
%F A000016 a(n) = 2^(n-1) - A327477(n). - _Gus Wiseman_, Sep 14 2019
%e A000016 For n=3 the 2 output sequences are 000111000111... and 010101...
%e A000016 For n=5 the 4 output sequences are those with periodic parts {0000011111, 0001011101, 0010011011, 01}.
%e A000016 For n=6 there are 6 such sequences.
%p A000016 A000016 := proc(n) local d, t; if n = 0 then return 1 else t := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t := t + NumberTheory:-Totient(d)* 2^(n/d)/(2*n) fi od; return t fi end:
%t A000016 a[0] = 1; a[n_] := Sum[Mod[k, 2] EulerPhi[k]*2^(n/k)/(2*n), {k, Divisors[n]}]; Table[a[n], {n, 0, 35}](* _Jean-François Alcover_, Feb 17 2012, after Pari *)
%o A000016 (PARI) a(n)=if(n<1,n >= 0,sumdiv(n,k,(k%2)*eulerphi(k)*2^(n/k))/(2*n));
%o A000016 (Haskell)
%o A000016 a000016 0 = 1
%o A000016 a000016 n = (`div` (2 * n)) $ sum $
%o A000016 zipWith (*) (map a000010 oddDivs) (map ((2 ^) . (div n)) $ oddDivs)
%o A000016 where oddDivs = a182469_row n
%o A000016 -- _Reinhard Zumkeller_, May 01 2012
%o A000016 (Python)
%o A000016 from sympy import totient, divisors
%o A000016 def A000016(n): return sum(totient(d)<<n//d-1 for d in divisors(n>>(~n&n-1).bit_length(),generator=True))//n if n else 1 # _Chai Wah Wu_, Feb 21 2023
%Y A000016 The main diagonal of table A068009, the left edge of triangle A053633.
%Y A000016 Cf. A000048, A000031, A000013, A053634, A182469.
%Y A000016 Subsets whose mean is an element are A065795.
%Y A000016 Dominated by A082550.
%Y A000016 Partitions containing their mean are A237984.
%Y A000016 Subsets containing n but not their mean are A327477.
%Y A000016 Cf. A051293, A135342, A240850, A327471, A327478.
%K A000016 nonn,nice,easy,changed
%O A000016 0,4
%A A000016 _N. J. A. Sloane_
%E A000016 More terms from _Michael Somos_, Dec 11 1999
| 997961f9010a013dd52359f9e992f347 |
A000017 | null | 7 | 1999-12-11T03:00:00 | [
"1",
"0",
"0",
"2",
"2",
"4",
"8",
"4",
"16",
"12",
"48",
"80",
"136",
"420",
"1240",
"2872",
"7652",
"18104",
"50184"
] | Erroneous version of A032522. | null | null | [
"dead"
] | 0 | 5 | oeisdata/seq/A000/A000017.seq | %I A000017 #7 Dec 11 1999 03:00:00
%S A000017 1,0,0,2,2,4,8,4,16,12,48,80,136,420,1240,2872,7652,18104,50184
%N A000017 Erroneous version of A032522.
%K A000017 dead
%O A000017 1,4
| 5d3e55c420e76bd3f14e73030fdae349 |
A000018 | [
"M0331",
"N0126"
] | 36 | 2023-07-08T01:17:54 | [
"1",
"1",
"2",
"2",
"4",
"8",
"13",
"25",
"44",
"83",
"152",
"286",
"538",
"1020",
"1942",
"3725",
"7145",
"13781",
"26627",
"51572",
"100099",
"194633",
"379037",
"739250",
"1443573",
"2822186",
"5522889",
"10818417",
"21209278",
"41613288",
"81705516",
"160532194",
"315604479",
"620834222",
"1221918604",
"2406183020",
"4740461247"
] | Number of positive integers <= 2^n of form x^2 + 16*y^2. | null | N. J. A. Sloane | [
"nonn"
] | 0 | 5 | oeisdata/seq/A000/A000018.seq | %I A000018 M0331 N0126 #36 Jul 08 2023 01:17:54
%S A000018 1,1,2,2,4,8,13,25,44,83,152,286,538,1020,1942,3725,7145,13781,26627,
%T A000018 51572,100099,194633,379037,739250,1443573,2822186,5522889,10818417,
%U A000018 21209278,41613288,81705516,160532194,315604479,620834222,1221918604,2406183020,4740461247
%N A000018 Number of positive integers <= 2^n of form x^2 + 16*y^2.
%D A000018 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000018 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000018 Delbert L. Johnson, <a href="/A000018/b000018.txt">Table of n, a(n) for n = 0..45</a>
%H A000018 D. Shanks and L. P. Schmid, <a href="http://dx.doi.org/10.1090/S0025-5718-1966-0210678-1">Variations on a theorem of Landau. Part I</a>, Math. Comp., 20 (1966), 551-569.
%H A000018 <a href="/index/Qua#quadpop">Index entries for sequences related to populations of quadratic forms</a>
%o A000018 (PARI) a(n)=local(A);if(n<0,0,A=qfrep([1,0;0,16],2^n);sum(k=1,2^n,A[k]!=0))
%K A000018 nonn
%O A000018 0,3
%A A000018 _N. J. A. Sloane_
%E A000018 More terms from _David W. Wilson_, Feb 07 2000
%E A000018 Definition corrected by _Sean A. Irvine_, Sep 09 2009
| 7ed0198d68c36a83f13c6aad0dcccdc7 |
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