A000638 -id:A000638 - 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

A000019 Number of primitive permutation groups of degree n.

Original entry on oeis.org

1, 1, 2, 2, 5, 4, 7, 7, 11, 9, 8, 6, 9, 4, 6, 22, 10, 4, 8, 4, 9, 4, 7, 5, 28, 7, 15, 14, 8, 4, 12, 7, 4, 2, 6, 22, 11, 4, 2, 8, 10, 4, 10, 4, 9, 2, 6, 4, 40, 9, 2, 3, 8, 4, 8, 9, 5, 2, 6, 9, 14, 4, 8, 74, 13, 7, 10, 7, 2, 2, 10, 4, 16, 4, 2, 2, 4, 6, 10, 4, 155, 10, 6, 6, 6, 2, 2, 2, 10, 4, 10, 2

Offset: 1

N. J. A. Sloane

A check found errors in Theißen's data (degree 121 and 125) as well as in Short's work (degree 169). - Alexander Hulpke, Feb 19 2002

There is an error at n=574 in the Dixon-Mortimer paper. - Colva M. Roney-Dougal.

CRC Handbook of Combinatorial Designs, 1996, pp. 595ff.
K. Harada and H. Yamaki, The irreducible subgroups of GL_n(2) with n <= 6, C. R. Math. Rep. Acad. Sci. Canada 1, 1979, 75-78.
A. Hulpke, Konstruktion transitiver Permutationsgruppen, Dissertation, RWTH Aachen, 1996.
M. W. Short, The Primitive Soluble Permutation Groups of Degree less than 256, LNM 1519, 1992, Springer
C. C. Sims, Computational methods in the study of permutation groups, pp. 169-183 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
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. Theißen, Eine Methode zur Normalisatorberechnung in Permutationsgruppen mit Anwendungen in der Konstruktion primitiver Gruppen, Dissertation, RWTH, RWTH-A, 1997 [But see comment above about errors! ]

Vaclav Kotesovec, Table of n, a(n) for n = 1..4095, computed using the Magma command shown below (terms 1..2499 from N. J. A. Sloane, computed using the GAP command shown below, which uses the results of Colva M. Roney-Dougal, a(1575) corrected).
Soleyman Askary, Nader Biranvand, and Farrokh Shirjian, New constructions of orbit codes based on imprimitive wreath products and wreathed tensor products, Rend. Circ. Mat. Palermo Ser. II (2023).
J. D. Dixon and B. Mortimer, The primitive permutation groups of degree less than 1000, Math. Proc. Cambridge Philos. Soc., 103, 213-238, 1988 [But see comment above about errors! ]
D. Holt, Enumerating subgroups of the symmetric group, in Computational Group Theory and the Theory of Groups, II, edited by L.-C. Kappe, A. Magidin and R. Morse. AMS Contemporary Mathematics book series, vol. 511, pp. 33-37. [Annotated copy]
A. Hulpke, Transitive groups of small degree
A. Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), 1-30.
J. Labelle and Y. N. Yeh, The relation between Burnside rings and combinatorial species, J. Combin. Theory, A 50 (1989), 269-284. See page 280.
C. C. Sims, Letter to N. J. A. Sloane (no date)
Index entries for sequences related to groups
Index entries for "core" sequences

Cf. A000001, A023675, A023676, A000638, A002106, A005432, A000637.

GAP
```
List([2..2499],NrPrimitiveGroups);
```

[NumberOfPrimitiveGroups(i) : i in [1..4095]];

More terms and additional references from Alexander Hulpke

A005432 Number of permutation groups of degree n (or, number of distinct subgroups of symmetric group S_n, counting conjugates as distinct).

Original entry on oeis.org

1, 1, 2, 6, 30, 156, 1455, 11300, 151221, 1694723, 29594446, 404126228, 10594925360, 175238308453, 5651774693595, 117053117995400, 5320744503742316, 125889331236297288, 7598016157515302757

Offset: 0

N. J. A. Sloane, Simon Plouffe

Labeled version of A000638.

L. Pyber shows c^(n^2(1+o(1))) <= a(n) <= d^(n^2(1+o(1))), c=2^(1/16), d=24^(1/6); conjectures lower bound is accurate.

C. C. Sims, Computational methods in the study of permutation groups, pp. 169-183 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Piotr Graczyk, Hideyuki Ishi, Kołodziejek Bartosz, Hélène Massam, Model selection in the space of Gaussian models invariant by symmetry, arXiv:2004.03503 [math.ST], 2020.
D. F. Holt, Enumerating subgroups of the symmetric group.
D. Holt, Enumerating subgroups of the symmetric group, in Computational Group Theory and the Theory of Groups, II, edited by L.-C. Kappe, A. Magidin and R. Morse. AMS Contemporary Mathematics book series, vol. 511, pp. 33-37. [Annotated copy]
J. Labelle and Y. N. Yeh, The relation between Burnside rings and combinatorial species, J. Combin. Theory, A 50 (1989), 269-284.
L. Naughton and G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8.
Götz Pfeiffer, Numbers of subgroups of various families of groups
L. Pyber, Enumerating Finite Groups of Given Order, Ann. Math. 137 (1993), 203-220.
N. J. A. Sloane, Transforms
Dashiell Stander, Qinan Yu, Honglu Fan, and Stella Biderman, Grokking Group Multiplication with Cosets, arXiv:2312.06581 [cs.LG], 2023. See footnote, p. 25.
Index entries for sequences related to groups

Cf. A000001, A000019, A000638.

List([2..5],n->Sum( List( ConjugacyClassesSubgroups( SymmetricGroup(n)), Size))); [Alexander Hulpke]

n := 5; &+[ Length(s):s in SubgroupLattice(Sym(n)) ];

Exponential transform of A116655. Binomial transform of A116693. - Christian G. Bower, Feb 23 2006

a(9) and a(10) from Alexander Hulpke, Dec 03 2004
More terms from a(11) and a(12) added by Christian G. Bower, Feb 23 2006 based on Goetz Pfeiffer's web page.
a(13) added by Liam Naughton, Jun 09 2011
a(14)-a(18) from Holt reference, Wouter Meeussen, Jul 02 2013

A000637 Number of fixed-point-free permutation groups of degree n.

Original entry on oeis.org

1, 0, 1, 2, 7, 8, 37, 40, 200, 258, 1039, 1501, 7629, 10109, 54322, 83975, 527036, 780193, 5808293

Offset: 0

N. J. A. Sloane

a(1) = 0 since the trivial group of degree 1 has a fixed point. One could also argue that one should set a(1) = 1 by convention.

G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863-911.
D. Holt, Enumerating subgroups of the symmetric group. Computational Group Theory and the Theory of Groups, II, edited by L.-C. Kappe, A. Magidin and R. Morse. AMS Contemporary Mathematics book series, vol. 511, pp. 33-37.
A. Hulpke, Konstruktion transitiver Permutationsgruppen, Dissertation, RWTH Aachen, 1996.
A. Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), 1-30.
A. Hulpke, Constructing Transitive Permutation Groups, in preparation
C. C. Sims, Computational methods in the study of permutation groups, pp. 169-183 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
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).

G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863-911. [Annotated scanned copy]
D. Holt, Enumerating subgroups of the symmetric group, in Computational Group Theory and the Theory of Groups, II, edited by L.-C. Kappe, A. Magidin and R. Morse. AMS Contemporary Mathematics book series, vol. 511, pp. 33-37. [Annotated copy]
A. Hulpke, Transitive groups of small degree
A. C. Lunn and J. K. Senior, Isomerism and Configuration, J. Physical Chem. 33 (7) 1929, 1027-1079.
A. C. Lunn and J. K. Senior, Isomerism and Configuration, J. Physical Chem. 33 (7) 1929, 1027-1079. [Annotated scan of page 1069 only]
C. C. Sims, Letter to N. J. A. Sloane (no date)
Index entries for sequences related to groups

Cf. A000001, A000019, A000638, A002106, A005432, A005226.

Cf. A000019, A002106. Unlabeled version of A116693.

a(n) = A000638(n) - A000638(n-1). - Christian G. Bower, Feb 23 2006

More terms from Alexander Hulpke
a(2) and a(10) corrected, a(11) and a(12) added by Christian G. Bower, Feb 23 2006
Terms a(13)-a(18) were computed by Derek Holt and contributed by Alexander Hulpke, Jul 30 2010, who comments that he has verified the terms up through a(16).
Edited by N. J. A. Sloane, Jul 30 2010, at the suggestion of Michael Somos

A005226 Number of atomic species of degree n; also number of connected permutation groups of degree n.

Original entry on oeis.org

0, 1, 1, 2, 6, 6, 27, 20, 130, 124, 598, 641, 4850, 4772, 35625, 46074, 389839, 487408, 4617554

Offset: 0

Simon Plouffe

An atomic species is one that is not the product of smaller species. - Christian G. Bower, Feb 23 2006

A permutation group is connected if it is not the direct product of smaller permutation groups. - Christian G. Bower, Feb 23 2006

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 147.
Jacques Labelle, Quelques espèces sur les ensembles de petite cardinalité, Ann. Sc. Math. Québec 9.1 (1985): 31-58.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. Decoste, G. Labelle & J. Labelle, Espèces sur les petites cardinalités Tableaux divers, Université du Québec à Montréal (octobre 1988), Unpublished.
Jacques Labelle, Quelques espèces sur les ensembles de petite cardinalité, Ann. Sc. Math. Québec 9.1 (1985): 31-58. (Annotated scanned copy of preprint)
J. Labelle and Y. N. Yeh, The relation between Burnside rings and combinatorial species, J. Combin. Theory, A 50 (1989), 269-284.
L. Naughton and G. Pfeiffer, Integer sequences realized by the subgroup pattern of the symmetric group, arXiv:1211.1911 [math.GR], 2012-2013 and J. Int. Seq. 16 (2013) #13.5.8.
N. J. A. Sloane, Transforms

Cf. A005227. Unlabeled version of A116655.

A000638 = Cases[Import["https://oeis.org/A000638/b000638.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{0} ~Join~ EulerInvTransform[A000638 // Rest] (* Jean-François Alcover, Dec 03 2019, updated Mar 17 2020 *)

Inverse Euler transform of A000638. Define b(n), c(n), d(): b(1)=d(1)=0. b(k)=A005227(k), k>1. c(k)=A000638(k), k>0. d(k)=a(k), k>1. d is Dirichlet convolution of b and c. - Christian G. Bower, Feb 23 2006

a(11) corrected and a(12) added by Christian G. Bower, Feb 23 2006 based on Goetz Pfeiffer's edit to A000638.
Could be extended to a(18) now using the new terms for A000637. - N. J. A. Sloane, Jul 30 2010
a(13) from Liam Naughton, Nov 23 2012
a(14)-a(18) from the inverse Euler transform of A000637. - R. J. Mathar, Mar 03 2015

A029726 Number of conjugacy classes of subgroups of the alternating group A_n.

Original entry on oeis.org

1, 1, 1, 2, 5, 9, 22, 40, 137, 223, 430, 788, 2537, 4558, 12136, 24871, 95526, 182234

Offset: 0

N. J. A. Sloane

Number of even permutation groups; also number of molecular species based on even permutation groups. - Christian G. Bower, Feb 23 2006

L. Naughton and G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, arXiv preprint arXiv:1211.1911, 2012 and J. Int. Seq. 16 (2013) #13.5.8
Götz Pfeiffer, Numbers of subgroups of various families of groups
N. J. A. Sloane, Transforms

Cf. A000001, A000638. Unlabeled version of A029725. Euler transform of A116653.

# GAP 4.2
Length(ConjugacyClassesSubgroups(AlternatingGroup(n)));

Magma
```
n := 5; #SubgroupLattice(Alt(n));
```

More terms from Reiner Martin, Dec 30 2001
a(10)-a(13) added by Christian G. Bower, Feb 23 2006 based on Goetz Pfeiffer's web page.
a(14)-a(17) from Vaclav Kotesovec, Jul 21 2022

A001051 Number of subgroups of order n in orthogonal group O(3).

Original entry on oeis.org

1, 3, 1, 5, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 10, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 7, 1, 5, 1, 8

Offset: 1

J. H. Conway

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Orthogonal Group.

The main sequences concerned with group theory are A000001, A000679, A001034, A001051, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A051881.

a[2] = 3; a[4] = 5; a[12] = 8; a[24] = 10; a[48] = a[60] = a[120] = 8; a[n_] := Switch[Mod[n, 4], 0, 7, 1, 1, 2, 5, 3, 1]; Table[a[n], {n, 1, 96}] (* Jean-François Alcover, Oct 15 2013 *)

A001051(n) = if((12==n)||(48==n)||(60==n)||(120==n),8,if(24==n,10,if((4==n)||(2==n),1+n,[1,5,1,7][1+((n-1)%4)]))); \\ Antti Karttunen, Jan 15 2019

Has period 1 5 1 7 except that a(2) = 3, a(4) = 5, a(12) = 8, a(24) = 10, a(48) = a(60) = a(120) = 8.

Data section extended up to a(120) by Antti Karttunen, Jan 15 2019

A005227 Number of atomic species of degree n which are not nontrivial substitutions.

Original entry on oeis.org

0, 1, 1, 2, 4, 6, 19, 20, 111, 116, 567, 641, 4718, 4772, 35489, 46012, 389277, 487408, 4616580

Offset: 0

N. J. A. Sloane

Labelle, Jacques. Quelques espèces sur les ensembles de petite cardinalité, Ann. Sc. Math. Québec 9.1 (1985): 31-58.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. Decoste, G. Labelle, & J. Labelle, Espèces sur les petites cardinalités Tableaux divers, Université du Québec à Montréal (octobre 1988), Unpublished.
Jacques Labelle, Quelques espèces sur les ensembles de petite cardinalité, Ann. Sc. Math. Québec 9.1 (1985): 31-58. (Annotated scanned copy of preprint)
J. Labelle and Y. N. Yeh, The relation between Burnside rings and combinatorial species, J. Combin. Theory, A 50 (1989), 269-284.

Cf. A005226.

Define b(n), c(n), d(n): b(1)=d(1)=0. b(k)=a(k), k>1. c(k)=A000638(k), k>0, d(k)=A005226(k), k>1. d is Dirichlet convolution of b and c. - Christian G. Bower, Feb 23 2006

a(8), a(11) corrected and a(12) added by Christian G. Bower, Feb 23 2006 based on Goetz Pfeiffer's edit to A000638

a(13)-a(18) from Vaclav Kotesovec, Jul 18 2022

A007649 Number of set-like molecular species of degree n.

Original entry on oeis.org

1, 1, 2, 3, 7, 9, 20, 26, 54, 74, 137, 184, 356, 473, 841, 1154, 2034, 2742, 4740, 6405, 10874, 14794, 24515, 33246, 54955, 74380, 120501, 163828, 263144, 356621, 567330, 768854, 1212354, 1644335, 2567636, 3478873, 5403223, 7314662, 11265825, 15258443, 23363143, 31608055, 48113280, 65063640, 98501538, 133168305, 200503864

Offset: 0

Simon Plouffe

nonn

A set-like molecular species is formed from the species of sets, E, through species addition, product and substitution.

G. Labelle and P. Leroux, Identities and enumeration: weighting connected components, Abstracts Amer. Math. Soc., 15 (1994), Meeting #896.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Joerg Arndt, Table of n, a(n) for n = 0..100
G. Labelle and P. Leroux, Identities and enumeration: weighting connected components, Abstracts Amer. Math. Soc., 15 (1994), Meeting #896. (Annotated scanned copy)
G. Labelle and P. Leroux, An extension of the exponential formula in enumerative combinatorics, The Electronic Journal of Combinatorics, Volume 3, Issue 2 (1996) (The Foata Festschrift volume), Research Paper #R12.
N. J. A. Sloane, Transforms.

Cf. A000638, A007650.

NN = 66; va = Array[0&, NN]; va[[1]] = 0; va[[2]] = 1; vm = Array[0&, NN]; vm[[1]] = 1; vm[[2]] = 1; For[n = 2, n <= NN - 1, n++, va[[n+1]] = DivisorSum[n , vm[[#+1]]&]; vm[[n+1]] = 1/n*Sum[DivisorSum[k, #*va[[#+1]] &]*vm[[n-k+1]], {k, 1, n}]]; vm (* Jean-François Alcover, Dec 01 2015, adapted from Joerg Arndt's PARI script *)

/* From the Labelle/Leroux reference */
N=66; /* that many terms */
va=vector(N);  va[0+1] = 0; va[1+1] = 1;
vm=vector(N);  vm[0+1] = 1; vm[1+1] = 1;
{ for(n=2, N-1,
    va[n+1] = sumdiv(n, k, vm[k+1]);
    vm[n+1] = 1/n * sum(k=1, n,  sumdiv(k, d, d*va[d+1]) * vm[n-k+1] );
); }
v007649=vm
/* v007650=va */
/* Joerg Arndt, Jul 30 2012 */

Euler Transform of A007650. Define c(n): c(0)=0. c(k)=a(k), k>0. A007650=MOEBIUSi(c)-c. - Christian G. Bower, Feb 23 2006

Added more terms, Joerg Arndt, Jul 30 2012

cp's OEIS Frontend

A000001 Number of groups of order n.

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A000019 Number of primitive permutation groups of degree n.

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A005432 Number of permutation groups of degree n (or, number of distinct subgroups of symmetric group S_n, counting conjugates as distinct).

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A000637 Number of fixed-point-free permutation groups of degree n.

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A005226 Number of atomic species of degree n; also number of connected permutation groups of degree n.

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A029726 Number of conjugacy classes of subgroups of the alternating group A_n.

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A001051 Number of subgroups of order n in orthogonal group O(3).

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