A000019 -id:A000019 - 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

A000638 Number of permutation groups of degree n; also number of conjugacy classes of subgroups of symmetric group S_n; also number of molecular species of degree n.

Original entry on oeis.org

1, 1, 2, 4, 11, 19, 56, 96, 296, 554, 1593, 3094, 10723, 20832, 75154, 159129, 686165, 1466358, 7274651

Offset: 0

N. J. A. Sloane

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 147.
Labelle, Jacques. "Quelques espèces sur les ensembles de petite cardinalité.", Ann. Sc. Math. Québec 9.1 (1985): 31-58.
G. Pfeiffer, Counting Transitive Relations, preprint 2004.
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. Decoste, G. Labelle, & J. Labelle, Espèces sur les petites cardinalités Tableaux divers, Université du Québec à Montréal (octobre 1988), Unpublished.
Justine Falque, On the enumeration of P-oligomorphic groups, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 25-26.
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]
Jacques Labelle, Quelques espèces sur les ensembles de petite cardinalité, Ann. Sc. Math. Québec 9.1 (1985): 31-58. (Annotated scanned copy of preprint) published volume
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]
L. Naughton and G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, arXiv preprint arXiv:1211.1911 [math.GR], 2012 and J. Int. Seq. 16 (2013) #13.5.8
Götz Pfeiffer, Numbers of subgroups of various families of groups
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Colin D. Reid, Simon M. Smith, Groups acting on trees with Tits' independence property (P), arXiv:2002.11766 [math.GR], 2020.
C. C. Sims, Letter to N. J. A. Sloane (no date)
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.
G. Xiao, PermGroup
Index entries for sequences related to groups

Partial sums of A000637.

Cf. A000001, A000019. Unlabeled version of A005432.

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

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

Euler Transform of A005226. Define b(n), c(n), d(n): b(1)=d(1)=0. b(k)=A005227(k), k>1. c(k)=a(k), k>0, d(k)=A005226(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 Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

Extended to a(18) using Derek Holt's data from A000637. - N. J. A. Sloane, Jul 31 2010

A002106 Number of transitive permutation groups of degree n.

Original entry on oeis.org

1, 1, 2, 5, 5, 16, 7, 50, 34, 45, 8, 301, 9, 63, 104, 1954, 10, 983, 8, 1117, 164, 59, 7, 25000, 211, 96, 2392, 1854, 8, 5712, 12, 2801324, 162, 115, 407, 121279, 11, 76, 306, 315842, 10, 9491, 10, 2113, 10923, 56, 6

Offset: 1

N. J. A. Sloane

It is conjectured that this is the number of Galois groups for irreducible polynomials of order n. (All such Galois groups are transitive.) - Charles R Greathouse IV, May 28 2014

Let G be a transitive permutation groups of degree n. Then G occurs as a Galois group for an irreducible polynomial of degree n with coefficients K if and only if K admits a Galois extension with Galois group G. ("=>" is true by definition of the Galois group for an irreducible polynomial; for "<=", see user631's answer in the Math Overflow link). Hence the conjecture above is equivalent to the inverse Galois problem. Every finite group can be realized as a Galois group of some extension L/K, but for a fixed base field K (for example, K = Q is the field of rational numbers) the question is usually open. - Jianing Song, May 26 2025

			a(3)=2: A_3 and S_3.

G. Butler and J. McKay, personal communication.
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).

Marina Anagnostopoulou-Merkouri, R. A. Bailey, and Peter J. Cameron, Permutation groups, partition lattices and block structures, arXiv:2409.10461 [math.GR], 2024. See Table 1 p. 45.
G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863-911.
G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863-911. [Annotated scanned copy]
John J. Cannon and Derek F. Hol, The transitive permutation groups of degree 32
F. N. Cole, Note on the substitution groups of six, seven, and eight letters, Bull. Amer. Math. Soc. 2 (1893), 184-190. Gives a(8)=48 instead of 50.
Computational Algebra Group, Summary of New Features in Magma V2.21
J. Conway, A. Hulpke, and J. McKay, On Transitive Permutation Groups, LMS Journal of Computation and Mathematics 1 (1998), pp. 1-8. See especially Appendix A.
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]
Derek Holt and Gordon Royle, A Census of Small Transitive Groups and Vertex-Transitive Graphs, arXiv:1811.09015 [math.CO], 2018.
A. Hulpke, Transitive groups of small degree
A. Hulpke, Konstruktion transitiver Permutationsgruppen, Dissertation, RWTH Aachen, 1996.
A. Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), 1-30.
E. G. Köhler, F. H. Lutz, Triangulated manifolds with few vertices: Vertex-transitive triangulations, arXiv:math/0506520 [math.GT], 2005.
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.
Math Overflow, What (permutation) groups can occur as galois groups of irreducible polynomials of degree n
G. A. Miller, On the lists of all the substitution groups that can be formed with a given number of elements, Bull. Amer. Math. Soc., 2 (1896), 138-145.
Wikipedia, Inverse Galois problem
Index entries for sequences related to groups
Index entries for "core" sequences

Cf. A000001, A000019, A177244, A186277.

a:=function(n)
return Length(AllTransitiveGroups(NrMovedPoints,n));
end; # Charles R Greathouse IV, May 28 2014

Corrected and extended to degree 31 by Alexander Hulpke, Aug 15 1996
Further corrections from Alexander Hulpke, Feb 19 2002
Degree 32 extended by Artur Jasinski, Feb 17 2011
Extended to degree 47 by Gabriel Verret, May 07 2016

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

A051070 a(n) is the n-th term in sequence A_n, respecting the offset, or a(n) = -1 if A_n has fewer than n terms.

Original entry on oeis.org

1, 2, 1, 0, 2, 3, 0, 7, 8, 4, 63, 1, 316, 78, 16, 2048, 7652, 26627, 8, 24000, 232919, 1145406, 3498690007594650042368, 2058537, 58, 26, 27, 59, 9272780, 3, 69273668, 4870847, 2387010102192469724605148123694256128, 1, 1, -53, 43, 0, -4696, 173, 44583, 111111111111111111111111111111111111111111, 30402457, 668803781, 1134903170, 382443020332

Offset: 1

Robert G. Wilson v, Aug 23 2000

a(58) = A000058(58) = 192523...920807 (58669977298272603 digits) is too large to include in the b-file. - Pontus von Brömssen, May 19 2022
Comment from N. J. A. Sloane, Dec 26 2022 (Start)
Note that a(n) = -1 can arise in two ways: either A_n has fewer than n terms, or A_n has at least n terms, but its n-th term is -1.
Here is a summary of the terms with n <= 80.
a(n) = -1 occurs just twice, for n = 53 and 54, in both cases because the relevant New York subway lines do not have enough stops.
a(1) though a(65) are known, although a(58) = = 192523...920807 has 58669977298272603 digits.
a(66) is the first unknown value.
Also unknown for n <= 80 are a(67), a(72), a(74), a(75), a(76), and a(77) (counts of numbers <= 2^n represented by various quadratic forms; some of these do not even have b-files), and a(80), which like a(66) is a graph-theory question. (End)

			a(19) = 8 because A000019(19) = 8.
a(20) = 24000 because A000020(20) = 24000.

Seth A. Troisi, Table of n, a(n) for n = 1..57 (terms 1..48 from Pontus von Brömssen).
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
Index entries for sequences whose definition involves A_n (or An).

See A091967, A107357, A102288 for other versions. See also A031214, A031135.

Cf. A000019, A000020, A000058.

for m from 1 do
  url:= sprintf("https://oeis.org/A%06d/b%06d.txt",m,m);
  S:= URL:-Get(url);
  L:= StringTools[Split](S,"\n");
  for t in L do
    g:= sscanf(t, "%d %d");
    if nops(g) = 2 and g[1] = m then
      a[m]:= g[2];
      break
    fi;
  od;
  if not assigned(a[m]) then break fi;
od:
seq(a[i],i=1..m-1); # Robert Israel, May 31 2015

Rechecked and 4 more terms added by N. J. A. Sloane, May 25 2005
a(36) and a(42) corrected and a(43) to a(46) added by Robert Israel, May 31 2015
Definition revised by N. J. A. Sloane, Nov 27 2016

A053169 A paradoxical sequence: a positive integer n is in this sequence if and only if n is not in sequence A_n in the database.

Original entry on oeis.org

4, 7, 9, 11, 12, 13, 15, 17, 18, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Miklos Szabo (mike(AT)ludens.elte.hu), N. J. A. Sloane, Feb 29 2000

"Not in sequence A_n" means not in the full list of terms, not simply in the list of terms visible in the entry.

The paradox is of course: is 53169 in this sequence?

			Sequence A000001 contains 1, so 1 is not in the sequence; A000002 contains 2, so 2 is not in the sequence; ...

Smullyan, Raymond M, What is the name of this book? : The riddle of Dracula and other logical puzzles, Englewood Cliffs, NJ : Prentice-Hall, c1978, see puzzle 163.

Index entries for sequences whose definition involves A_n (or An).

Cf. A107357.

Complement of A053873.

Thanks to Alexander Hulpke for the information that 19 is in A000019.

Extended to n=100 by N. J. A. Sloane, May 26 2007

A023676 Table of orders of transitive permutation groups by degree.

Original entry on oeis.org

3, 6, 4, 4, 8, 12, 24, 5, 10, 20, 60, 120, 6, 6, 12, 12, 18, 24, 24, 24, 36, 36, 48, 60, 72, 120, 360, 720, 7, 14, 21, 42, 168, 2520, 5040

Offset: 3

N. J. A. Sloane

			{3,6}, {4,4,8,12,24}, ...

M. Pohst and H. Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989. Appendix: Numerical Tables, p. 430.

Index entries for sequences related to groups

Cf. A000001, A000019, A023675.

A102842 Insipid numbers: n is defined to be insipid if "G is a primitive subgroup of the symmetric group S_n" implies that "G=A_n or G=S_n".

Original entry on oeis.org

1, 2, 3, 4, 34, 39, 46, 51, 58, 69, 70, 75, 76, 86, 87, 88, 92, 93, 94, 95, 96, 99, 106, 111, 115, 116, 118, 123, 124, 134, 141, 142, 143, 145, 146, 147, 148, 154, 159, 160, 161, 166, 172, 177, 178, 184, 185, 187, 188, 189, 195, 201, 202, 204, 205, 206, 207, 209

Offset: 1

David L. Harden, Feb 27 2005

nonn

A few basic properties: No prime p > 3 is in this sequence, since the subgroup of S_p generated by any p-cycle is primitive (and too small to be A_p or S_p when p>3).
It seems hard to find long gaps in this sequence. It seems plausible (this is implied by some conjectures in number theory) that there are infinitely many strings of 5 consecutive positive integers not in this sequence; however, I do not know of a construction which should yield infinitely many strings of 6 consecutive positive integers which are in the sequence (this may be just a reflection of my ignorance of the right families of finite groups); the largest example I know of a string of more than 5 consecutive integers not in this sequence has length 7 and first term 2^150-5.
If q is a power of a prime and d > 1 is a positive integer (except in the cases where d=2 and q <= 4, in which this construction yields symmetric or alternating groups), then (q^d-1)/(q-1) is not insipid for the following reason:
The group PGL(d,q) acts doubly transitively (and therefore primitively) on the (q^d-1)/(q-1) 1-dimensional subspaces of a d-dimensional vector space over the finite field of order q. In particular, when d=2, this number is q+1 and this is why each power of a prime (including the primes themselves) prevents the next positive integer from being insipid, in addition to being noninsipid itself. This is why the explanation for why 38 is noninsipid just said that 38=37+1.
The Magma code generates the insipid numbers <= U, with the exceptions of 1 and 2. Since I do not know Magma well enough to judge this for myself, it is possible that U has to be a constant (and not just another program variable) for this code to work properly.
This is the set of n such that n = 1 or 2 and A000019(n)=2.
The link gives all the insipid numbers < 1000, except for 1 and 2. - David L. Harden, Aug 15 2007
There are infinitely many insipid numbers. In fact, they are of density 1, because P. J. Cameron, P. M. Neumann and D. N. Teague proved that the number of non-insipid numbers less than n grows like 2n/log(n). - Sébastien Palcoux, Jul 23 2019

			39 is the next term after 34 because it is possible to construct primitive nonnormal subgroups of S_n for n=35,36,37 and 38:
35: 35=(7 3) and 3 < 7/2 so S_7 acts primitively on 35 points because S_7 has maximal subgroups isomorphic to S_3 x S_4.
36: 36=(9 2) and 2 < 9/2 so S_9 acts primitively on 36 points because S_9 has maximal subgroups isomorphic to S_2 x S_7.
37: 37 is prime.
38: 38=37+1.

J. Dixon and B. Mortimer: Permutation groups. Springer 1996, 360pp.

David L. Harden, Table of n, a(n) for n = 1..486 (insipid numbers below 1000) [Corrected Aug 25 2007]
P. J. Cameron, P. M. Neumann, D. N. Teague, On the degrees of primitive permutation groups, Math. Z. 180 (1982), 141-149.
S. Palcoux, Are there infinitely many insipid numbers? (version: 2019-07-22), MathOverflow.
C. M. Roney-Dougal, The primitive permutation groups of degree less than 2500, Journal of Algebra 292 (2005) 154-183.

Cf. A000019.

[n : n in [1..U] | NumberOfPrimitiveGroups(n) eq 2];

cp's OEIS Frontend

A173397 Partial sums of A000019.

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A000001 Number of groups of order n.

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A000638 Number of permutation groups of degree n; also number of conjugacy classes of subgroups of symmetric group S_n; also number of molecular species of degree n.

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A002106 Number of transitive 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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A051070 a(n) is the n-th term in sequence A_n, respecting the offset, or a(n) = -1 if A_n has fewer than n terms.

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