A046057 -id:A046057 - cp's OEIS frontend

A000001 Number of groups of order n.

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, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 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

Offset: 0

N. J. A. Sloane

Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - Lekraj Beedassy, Dec 16 2004
Also, number of nonisomorphic primitives (antiderivatives) of the combinatorial species Lin[n-1], or X^{n-1}; see Rajan, Summary item (i). - Nicolae Boicu, Apr 29 2011
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
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
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
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

			Groups of orders 1 through 10 (C_n = cyclic, D_n = dihedral of order n, Q_8 = quaternion, S_n = symmetric):
1: C_1
2: C_2
3: C_3
4: C_4, C_2 X C_2
5: C_5
6: C_6, S_3=D_6
7: C_7
8: C_8, C_4 X C_2, C_2 X C_2 X C_2, D_8, Q_8
9: C_9, C_3 X C_3
10: C_10, D_10

S. R. Blackburn, P. M. Neumann, and G. Venkataraman, Enumeration of Finite Groups, Cambridge, 2007.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 302, #35.
J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 150.
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.
M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
D. Joyner, 'Adventures in Group Theory', Johns Hopkins Press. Pp. 169-172 has table of groups of orders < 26.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.
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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. U. Besche and Ivan Panchenko, Table of n, a(n) for n = 0..2047 [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. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927).
Hans Ulrich Besche and Bettina Eick, Construction of finite groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
Hans Ulrich Besche and Bettina Eick, The groups of order at most 1000 except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
H. U. Besche, B. Eick and E. A. O'Brien, The groups of order at most 2000, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.
H. U. Besche, B. Eick, E. A. O'Brien and Max Horn, The Small Groups Library
H. U. Besche, B. Eick and E. A. O'Brien, Number of isomorphism types of finite groups of given order [gives incorrect a(1024)]
H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
Henry Bottomley, Illustration of initial terms
David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3.
J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.
Marek Dančo, Mikoláš Janota, Michael Codish, and João Jorge Araújo, Complete Symmetry Breaking for Finite Models, arXiv:2502.10155 [cs.LO], 2025. See p. 7. Mentions this sequence.
Yang-Hui He and Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019.
Otto Hölder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 43 pp. 301-412 (1893).
Max Horn, Numbers of isomorphism types of finite groups of given order
Rodney James, The groups of order p^6 (p an odd prime), Math. Comp. 34 (1980), 613-637.
Rodney James and John Cannon, Computation of isomorphism classes of p-groups, Mathematics of Computation 23.105 (1969): 135-140.
Olexandr Konovalov, Crowdsourcing project for the database of numbers of isomorphism types of finite groups, Github (a list of gnu(n) for many n < 50000).
Desmond MacHale, Are There More Finite Rings than Finite Groups?, Amer. Math. Monthly (2020) Vol. 127, Issue 10, 936-938.
Mehdi Makhul, Josef Schicho, and Audie Warren, On Galois groups of type-1 minimally rigid graphs, arXiv:2306.04392 [math.CO], 2023.
G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Dayanand S. Rajan, The equations D^kY=X^n in combinatorial species, Discrete Mathematics 118 (1993) 197-206 North-Holland.
E. Rodemich, The groups of order 128, J. Algebra 67 (1980), no. 1, 129-142.
Gordon Royle, Combinatorial Catalogues. See subpage "Generators of small groups" for explicit generators for most groups of even order < 1000. [broken link]
D. Rusin, Asymptotics [Cached copy of lost web page]
Eric Weisstein's World of Mathematics, Finite Group
Wikipedia, Finite group
M. Wild, The groups of order sixteen made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
Gang Xiao, SmallGroup
Index entries for sequences related to groups
Index entries for "core" sequences

The main sequences concerned with group theory are A000001 (this one), A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A000688, A060689, A051532.
Cf. A046058, A046059, A023675, A023676.
A003277 gives n for which A000001(n) = 1, A063756 (partial sums).
A046057 gives first occurrence of each k.
A027623 gives the number of rings of order n.

A000001 := Concatenation([0], List([1..500], n -> NumberSmallGroups(n))); # Muniru A Asiru, Oct 15 2017

D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] ]; // John Cannon, Dec 23 2006

GroupTheory:-NumGroups(n); # with(GroupTheory); loads this command - N. J. A. Sloane, Dec 28 2017

FiniteGroupCount[Range[100]] (* Harvey P. Dale, Jan 29 2013 *)
a[ n_] := If[ n < 1, 0, FiniteGroupCount @ n]; (* Michael Somos, May 28 2014 *)

From Mitch Harris, Oct 25 2006: (Start)
For p, q, r primes:
a(p) = 1, a(p^2) = 2, a(p^3) = 5, a(p^4) = 14, if p = 2, otherwise 15.
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.
a(p^e) ~ p^((2/27)e^3 + O(e^(8/3))).
a(p*q) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p. (p < q)
a(p*q^2) is one of the following:
   ---------------------------------------------------------------------------
  | a(p*q^2) |            p*q^2 of the form             |  Sequences (p*q^2)  |
   ---------- ------------------------------------------ ---------------------
  | (p+9)/2  |          q == 1 (mod p), p odd           |      A350638        |
  |    5     |         p=3, q=2 =>  p*q^2 = 12          |Special case with A_4|
  |    5     |               p=2, q odd                 |      A143928        |
  |    5     |             p == 1 (mod q^2)             |      A350115        |
  |    4     | p == 1 (mod q), p > 3, p !== 1 (mod q^2) |      A349495        |
  |    3     |       q == -1 (mod p), p and q odd       |      A350245        |
  |    2     |  q !== +-1 (mod p) and p !== 1 (mod q)   |      A350422        |
   ---------------------------------------------------------------------------
[Table from Bernard Schott, Jan 18 2022]
a(p*q*r) (p < q < r) is one of the following:
  q == 1 (mod p)  r == 1 (mod p)  r == 1 (mod q)  a(p*q*r)
  --------------  --------------  --------------  --------
        No              No              No            1
        No              No              Yes           2
        No              Yes             No            2
        No              Yes             Yes           4
        Yes             No              No            2
        Yes             No              Yes           3
        Yes             Yes             No           p+2
        Yes             Yes             Yes          p+4
[table from Derek Holt].
(End)
a(n) = A000688(n) + A060689(n). - R. J. Mathar, Mar 14 2015

More terms from Michael Somos

Typo in b-file description fixed by David Applegate, Sep 05 2009

A046056 Smallest order for which there are n nonisomorphic finite Abelian groups, or 0 if no such order exists.

Original entry on oeis.org

1, 4, 8, 36, 16, 72, 32, 900, 216, 144, 64, 1800, 0, 288, 128, 44100, 0, 5400, 0, 3600, 864, 256, 0, 88200, 1296, 0, 27000, 7200, 0, 512, 0, 5336100, 1728, 0, 2592, 264600, 0, 0, 0, 176400, 0, 1024, 0, 2304, 3456, 0, 0, 10672200, 7776, 32400, 0, 0, 0, 1323000, 5184, 2048, 0, 0, 0, 4608

Offset: 1

Eric W. Weisstein

There is a k: A000688(k)=n if and only if n is product of partition numbers.

Charlie Neder, Table of n, a(n) for n = 1..1000
B. Horvat, G. Jaklic and T. Pisanski, On the number of Hamiltonian groups, arXiv:math/0503183 [math.CO], 2005.
Charlie Neder, Python program for computing this sequence.
Eric Weisstein's World of Mathematics, Abelian Group.
Index entries for sequences related to groups

Cf. A000041, A000688, A033637, A046054, A046055, A046057.

More terms from Christian G. Bower

a(20) and a(28) corrected, and a(52)-a(60) from Charlie Neder, Jan 17 2019

A046058 Incrementally largest numbers of nonisomorphic finite groups of order n.

Original entry on oeis.org

1, 2, 5, 14, 15, 51, 52, 267, 2328, 56092, 10494213, 49487367289

Offset: 1

Eric W. Weisstein

H. U. Besche, The Small Groups library
Hans Ulrich Besche and Bettina Eick, Construction of finite groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
Hans Ulrich Besche and Bettina Eick, The groups of order at most 1000 except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
H. U. Besche, B. Eick and E. A. O'Brien, The groups of order at most 2000, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.
David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3.
Bettina Eick and E. A. O'Brien, Enumerating p-groups. Group theory. J. Austral. Math. Soc. Ser. A 67 (1999), no. 2, 191-205.
Eric Weisstein's World of Mathematics, Finite Group.
Index entries for sequences related to groups

Cf. A000001, A000679, A000688, A046057, A046059.

a(n) = A000001(A046059(n)). - M. F. Hasler, Feb 24 2015

a(11) and a(12) from Eamonn O'Brien, Apr 15 2002

a(12) corrected by David Burrell, Jun 06 2022

A053403 Consider the set P of pairs (a,b) generated by the rules: (1,1) is in P; if (a,b) is in P then (b,a+b) is in P; if (a,b) and (a',b') are in P then (aa', bb') is in P. Sequence gives numbers not appearing in P.

Original entry on oeis.org

7, 11, 19, 29, 31, 47, 49, 53, 67, 71, 73, 79, 87, 91, 103, 119, 127, 131, 137, 139, 141, 142, 143, 146, 147, 151, 155, 179, 191, 193, 201, 203, 211, 213, 219, 223, 227, 229, 235, 237, 239, 247, 251, 265, 271, 301, 329, 331, 337, 341, 343, 347, 355, 358, 359

Offset: 1

R. Keith Dennis (dennis(AT)math.cornell.edu), Jan 07 2000

Sequence has 508 known terms, the largest of which is 55487. Conjecturally it is finite. If it is and 55487 is the largest term, then the function the number of groups of order n takes on all positive integers as values.

			The pairs with b <= 7 are (1,1), (1,2), (1,4), (2,3), (2,6), (3,5), and (4,5).  Since none of these has b = 7, 7 can never appear in P. - _Charlie Neder_, Feb 01 2019

R. Keith Dennis, The number of groups of order n, Cambridge Tracts in Mathematics, number 173.
Claudia A. Spiro, Local distribution results for the group-counting function at positive integers. In Proceedings of the Sundance conference on combinatorics and related topics (Sundance, Utah, 1985). Congr. Numer. 50 (1985), 107-110. MR0833542 (87g:11117).

Charlie Neder, Table of n, a(n) for n = 1..508
Claudia Spiro, A Conjecture in Statistical Group theory, Blog Entry, Dec 26 2011.
Claudia Spiro, A Conjecture in Statistical Group theory, Blog Entry, Dec 26 2011 [Cached copy, permission requested]
Claudia A. Spiro-Silverman, When the group-counting function assumes a prescribed integer value at squarefree integers frequently, but not extremely frequently, Acta Arithmetica, 1992 | 61 | 1 | 1-12.
Index entries for sequences related to groups

Cf. A000001, A046057.

A240007 Smallest k such that the number of groups of order k is equal to prime(n), or 0 if no such k exists.

Original entry on oeis.org

4, 75, 8, 375, 140, 56, 675, 1029, 294, 1380, 0, 180, 420, 112, 120, 656, 6875, 312, 243, 3660, 0, 3612, 0, 4140, 6498, 0, 0, 0, 0, 810, 0, 1260, 792, 0, 0, 0, 0, 0, 1936, 0, 1456, 1320, 0, 0, 144, 1000, 1368, 0, 1404, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Michel Lagneau, Mar 30 2014

nonn

Smallest k such that A000001(k) = prime(n), or 0 if no such k exists.
It seems that there is no order for which the number of groups is 31, 59, 71, 73, 79, 83, 89, 97, 101, 103, 109, 127, 139,...
Above comment is incorrect. According to the Conway article, every n <= 10000000 is the number of groups of order k for some k. So all the 0 entries above are wrong, but we do not necessarily know the true value. - Eric M. Schmidt, Sep 14 2014

			a(6)= 56 because prime(6) = 13 => there exists 13 groups of order 56.

John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, The Mathematical Intelligencer, Spring 2008, Volume 30, Issue 2, pp 6-15, doi:10.1007/BF02985731.

Cf. A000001, A046057, A053403.

lst={};Do[k=1;While[!FiniteGroupCount[k]==Prime[n],k++];If[k==2048,AppendTo[lst,0],AppendTo[lst,k]],{n,1,70}];lst

Values for 59, 71, 79, 89, and 97 filled in from Conway link by Eric M. Schmidt, Sep 14 2014

A384146 Smallest squarefree order m > 0 for which there are n nonisomorphic finite groups of order m, or 0 if no such order exists.

Original entry on oeis.org

1, 6, 609, 30, 273, 42, 903, 510, 8729, 3255, 494711, 210, 16951, 5115, 54431, 1218

Offset: 1

Robin Jones, May 21 2025

It has been established that every n < 10000000 arises as the number of groups up to isomorphism of some squarefree m. That is, a(n) > 0 for n < 10000000.

It is conjectured that 0 never appears in this sequence.

			a(3)=609 since there are 3 groups of order 609 up to isomorphism, and 609 is the smallest squarefree integer such that there are 3 groups of that order.

J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.

Cf. A046057 (m not necessarily squarefree).

cp's OEIS Frontend

A000001 Number of groups of order n.

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A046056 Smallest order for which there are n nonisomorphic finite Abelian groups, or 0 if no such order exists.

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A046058 Incrementally largest numbers of nonisomorphic finite groups of order n.

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A053403 Consider the set P of pairs (a,b) generated by the rules: (1,1) is in P; if (a,b) is in P then (b,a+b) is in P; if (a,b) and (a',b') are in P then (aa', bb') is in P. Sequence gives numbers not appearing in P.

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A240007 Smallest k such that the number of groups of order k is equal to prime(n), or 0 if no such k exists.

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Programs

Mathematica

Extensions

A384146 Smallest squarefree order m > 0 for which there are n nonisomorphic finite groups of order m, or 0 if no such order exists.

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