A027623 -id:A027623 - 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

A037291 Number of rings with 1 containing n elements.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 11, 4, 1, 1, 4, 1, 1, 1, 50, 1, 4, 1, 4, 1, 1, 1, 11, 4, 1, 12, 4, 1, 1, 1, 208, 1, 1, 1, 16, 1, 1, 1, 11, 1, 1, 1, 4, 4, 1, 1, 50, 4, 4, 1, 4, 1, 12, 1, 11, 1, 1, 1, 4, 1, 1, 4

Offset: 1

Christian G. Bower, Jun 15 1998

Many authors simply call these "rings". They are also known as unital rings, rings with unity, or rings with identity. - Charles R Greathouse IV, Aug 12 2015

Is this sequence multiplicative? That is, if p and q are distinct primes, is it true that a(p^i*q^j) = a(p^i)*a(q^j)? - Jianing Song, Oct 26 2019. The answer is yes - see the Eric M. Rains link. - N. J. A. Sloane, Oct 27 2019

Gérard P. Michon, Ring Theory, Numericana, 2000-2022.
C. Noebauer, Home page and Table of numbers of small rings [Archived copies]
Eric M. Rains, The number of unital rings with n elements is a multiplicative function of n.

Cf. A027623, A037221, A127707.

a(16) and a(32)-a(63) from Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000
Keyword 'mult' added by Jianing Song, Feb 02 2020
a(54) corrected by Andrey Zabolotskiy, Feb 02 2023

A037289 Number of commutative rings with n elements.

Original entry on oeis.org

1, 2, 2, 9, 2, 4, 2, 34, 9, 4, 2, 18, 2, 4, 4, 162, 2, 18, 2, 18, 4, 4, 2, 68, 9, 4, 36, 18, 2, 8, 2

Offset: 1

Christian G. Bower, Jun 15 1998

These rings do not necessarily contain an identity element.

This sequence is multiplicative. See the reference "The Numbers of Small Rings" below, which proves the result for all rings; restricting to commutative rings only makes the proof easier. - Conjecture by Mitch Harris, Apr 19 2005, proof found by Franklin T. Adams-Watters, Jul 10 2012

Simon R. Blackburn and K. Robin McLean, The enumeration of finite rings, 2022 preprint. arXiv:2107.13215 [math.CO]
A. V. Lelechenko, Parity of the number of primes in a given interval and algorithms of the sublinear summation, arXiv preprint arXiv:1305.1639, 2013
C. Noebauer, Home page [Archived copy as of 2008 from web.archive.org]
Christof Noebauer, The numbers of small rings (PostScript).
C. Noebauer, Thesis on the enumeration of near-rings
Bjorn Poonen, The moduli space of commutative algebras of finite rank, J. Eur. Math. Soc. (JEMS) 10:3 (2008), pp. 817-836. arXiv:0608491 [This contains an error, see Blackburn & McLean]

Cf. A027623, A037291.

a(p^n) = p^(2/27 * n^3 + O(n^2.5)), see Blackburn & McLean. - Charles R Greathouse IV, Jul 13 2022

a(16) from Christof Noebauer (christof.noebauer(AT)algebra.uni-linz.ac.at), Sep 29 2000, who reports that the sequence continues a(32) = ? (> 876), a(33) = 4, 4, 4, 81, 2, 4, 4, 68, 2, 8, 2, 18, 18, 4, 2, 324, 9, 18, 4, 18, 2, 72, 4, 68, 4, 4, 2, 36, 2, 4, 18 = a(63), a(64) = ? (> 12696)

A127707 Number of commutative rings with 1 containing n elements.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 10, 4, 1, 1, 4, 1, 1, 1, 37, 1, 4, 1, 4, 1, 1, 1, 10, 4, 1, 11, 4, 1, 1, 1, 109, 1, 1, 1, 16, 1, 1, 1, 10, 1, 1, 1, 4, 4, 1, 1, 37, 4, 4, 1, 4, 1, 11, 1, 10, 1, 1, 1, 4, 1, 1, 4

Offset: 1

Hugues Randriam (randriam(AT)enst.fr), Jan 24 2007

Is this a multiplicative function?

Answer: yes! See the Eric M. Rains link for a proof for the result for all unital rings; restricting to commutative rings does not affect the essence of the proof. - Jianing Song, Feb 02 2020

C. Noebauer, Home page and Table of numbers of small rings [Archived copies from web.archive.org]
Eric M. Rains, The number of unital rings with n elements is a multiplicative function of n.

Cf. A037289, A037291, A127708, A027623, A307000, A341202.

a(n) = A037291(n) - A127708(n). - Bernard Schott, Apr 19 2022

Keyword 'mult' added by Jianing Song, Feb 02 2020
a(32)-a(63) using Nöbauer's data added by Andrey Zabolotskiy, Apr 18 2022
a(32) = 109 corrected by Bernard Schott, Apr 19 2022

A037234 a(n) = number of rings with n elements.

Original entry on oeis.org

0, 1, 2, 2, 11, 2, 4, 2, 52, 11, 4, 2, 22, 2, 4, 4, 390, 2, 22, 2, 22, 4, 4, 2, 104, 11, 4, 59, 22, 2, 8, 2

Offset: 0

N. J. A. Sloane

From M. F. Hasler, Jan 05 2021 (Start)
This entry complements the "main entry" A027623 which for all n >= 1 also gives the number of rings with n elements, but which has A027623(0) = 1 by explicit definition. (There is no ring with no elements, since a ring is an abelian group and therefore must have at least the 0 element.)
a(32) is presently unknown: see A027623 for lower bounds and values a(n) for n > 32. (End)

			From _Bernard Schott_, Jan 06 2021: (Start)
a(1) = 1: The ring with only one element, 0, is called the zero ring.
a(2) = 2: These two rings of order 2 with elements {0, a} form an abelian group for operator +: 0+0 = 0, 0+a = a+0 = a, a+a = 0.
   - The first ring is obtained for  multiplication defined by: 0*0 = 0*a = a*0 = 0, a*a = a. This ring is isomorphic to the field Z/2Z.
   - The second ring is given for 0*0 = 0*a = a*0 = a*a = 0. Here a is a divisor of 0. (End)

V. G. Antipkin and V. P. Elizarov, Rings of order p^3, Sib. Math. J. vol 23 no 4 (1982) pp 457-464, MR0668331 (84d:16025), doi:10.1007/BF00968650.
Eric Weisstein's World of Mathematics, Ring.

A027623 is the main entry for this sequence.

apply( A037234(n, e=0)=if( !e, vecprod([call(self(),f) | f <- factor(n)~]), e<3, [if(n,2), 11][e], e==3, if(n>2,3*sqrtnint(n,3),2)+50, n>2 || e>4, /*error*/("not yet implemented"), 390), [0..63]) \\ M. F. Hasler, Jan 05 2021

From M. F. Hasler, Jan 05 2021: (Start)
a(p) = 2 for any prime p.
a(m n) = a(m) a(n) when gcd(m,n) = 1. (Multiplicativity.)
a(p^2) = 11 for any prime p.
a(p^3) = 3p + 50 for any odd prime p [Antipkin & Elizarov]. (End)

A038538 Number of semisimple rings with n elements.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 13, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 6, 6, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it)

Enumeration uses Wedderburn-Artin theorem and fact that a finite division ring is a field.

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, pp. 274-276.
John Knopfmacher, Abstract analytic number theory, North-Holland, 1975, pp. 63-64.
T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, 2001.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Catalina Calderón and María José Zárate, The Number of Semisimple Rings of Order at most x, Extracta mathematicae, Vol. 7, No. 2-3 (1992), pp. 144-147.
J. Duttlinger, Eine Bemerkung zu einer asymptotischen Formel von Herrn Knopfmacher, Journal für die reine und angewandte Mathematik, Vol. 1974, No. 266 (1974), pp. 104-106.
John Knopfmacher, Arithmetical properties of finite rings and algebras, and analytic number theory, Journal für die reine und angewandte Mathematik, Volume 1972, No. 252 (1972), pp. 16-43.
Werner Georg Nowak, On the value distribution of a class of arithmetic functions, Commentationes Mathematicae Universitatis Carolinae, Vol. 37, No. 1 (1996), pp. 117-134.
Index entries for sequences computed from exponents in factorization of n.

Cf. A002110, A004101, A005117, A025487, A027623, A052305, A123030 (partial sums), A244285.

With[{emax = 7}, f[e_] := f[e] = Coefficient[Series[Product[1/(1 - x^(j*k^2)), {k, 1, Floor[Sqrt[emax]] + 1}, {j, 1, Floor[emax/k^2] + 1}], {x, 0, emax}], x, e]; a[1] = 1; a[n_] := Times @@ f /@ FactorInteger[n][[;; , 2]]; Array[a, 2^emax]] (* Amiram Eldar, Jan 31 2024, using code by Vaclav Kotesovec at A004101 *)

v004101from1 = [1, 2, 3, 6, 8, 13, 18, 29, 40, 58, 79, 115, 154, 213, 284, 391, 514, 690, 900, 1197]; \\ From the data-section of A004101.
A004101(n) = v004101from1[n];
vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
A038538(n) = vecproduct(apply(e -> A004101(e), factorint(n)[, 2])); \\ Antti Karttunen, Nov 18 2017

Multiplicative with a(p^k) = A004101(k).
For all n, a(A002110(n)) = a(A005117(n)) = 1.
From Amiram Eldar, Jan 31 2024: (Start)
Dirichlet g.f.: Product_{k,m>=1} zeta(k*m^2*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2.499616... = A244285 (see A123030 for a more precise asymptotic formula). (End)

cp's OEIS Frontend

A000001 Number of groups of order n.

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A037291 Number of rings with 1 containing n elements.

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A037289 Number of commutative rings with n elements.

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A127707 Number of commutative rings with 1 containing n elements.

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A037234 a(n) = number of rings with n elements.

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A038538 Number of semisimple rings with n elements.

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A127708 Number of non-commutative rings with 1 containing n elements.

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A037290 Number of self-converse rings (isomorphic to inverse) with n elements.

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