A000001 -id:A000001 - cp's OEIS frontend

A249555 Numbers m such that there are precisely 12 groups of order m.

88, 152, 184, 196, 204, 210, 248, 330, 344, 348, 376, 390, 462, 472, 484, 492, 536, 568, 570, 632, 636, 664, 714, 770, 824, 856, 858, 966, 1016, 1048, 1068, 1110, 1112, 1208, 1212, 1230, 1254, 1290, 1304, 1326, 1336, 1356, 1430, 1432, 1444, 1518, 1528, 1592, 1644

Offset: 1

N. J. A. Sloane, Nov 01 2014

nonn

Jorge R. F. F. Lopes, Table of n, a(n) for n = 1..1512
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), this sequence (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A249555 := Filtered([1..2015], n -> NumberSmallGroups(n) = 12); # Muniru A Asiru, Oct 16 2017

Select[Range@ 2074, FiniteGroupCount@ # == 12 &] (* Michael De Vlieger, Oct 16 2017. Note: extending the range to 2075 and further will result in incorrect output. - Andrey Zabolotskiy, Oct 27 2017 *)

A292896 Numbers m such that there are precisely 13 groups of order m.

Original entry on oeis.org

56, 60, 150, 189, 441, 726, 837, 945, 1012, 1161, 1204, 1521, 1575, 1647, 1734, 1809, 1988, 2079, 2133, 2205, 2366, 2619, 2781, 2925, 2948, 3174, 3213, 3556, 3610, 3753, 4077, 4239, 4324, 4347, 4851, 5046, 5211, 5697, 5805, 5908, 6021, 6183, 6507, 6692, 7479, 7497, 7605, 7623, 7641, 7749, 8410, 8451

Offset: 1

Muniru A Asiru, Oct 23 2017

nonn

			The 13 groups of order 56 have the following structure C7 : C8, C56, C7 : Q8, C4 x D14, D56, C2 x (C7 : C4), (C14 x C2) : C2, C28 x C2, C7 x D8, C7 x Q8, (C2 x C2 x C2) : C7, C2 x C2 x D14, C14 x C2 x C2 where C, D and Q mean Cyclic group, Dihedral group and Quarternion group of the stated order. The symbols x and : mean direct and semidirect products respectively.

Muniru A Asiru, Table of n, a(n) for n = 1..273
Gordon Royle, Numbers of Small Groups
H. U. Besche, B. Eick and E. A. O'Brien. The Small Groups Library
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), this sequence (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A292896 := Filtered([1..2015], n -> NumberSmallGroups(n) = 13);

More terms from Muniru A Asiru, Nov 18 2017

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

A249550 Numbers m such that there are precisely 7 groups of order m.

Original entry on oeis.org

375, 605, 903, 1705, 2255, 2601, 2667, 3081, 3355, 3905, 3993, 4235, 4431, 4515, 4805, 5555, 6123, 6355, 6375, 6765, 7077, 7205, 7865, 7917, 7959, 8305, 8405, 8625, 8841, 9455, 9723, 9933, 9955, 10285, 10505, 10875, 11005, 11487, 11495, 11571, 11605, 11715, 11935, 12207, 12505, 13005, 13053, 13251, 13255, 13335, 13805, 14133

Offset: 1

N. J. A. Sloane, Nov 01 2014

nonn

			For m = 375, the 7 groups are C375, ((C5 x C5) : C5) : C3, C75 x C5, C3 x ((C5 x C5) : C5), C3 x (C25 : C5), C5 x ((C5 x C5) : C3), C15 x C5 x C5 and for n = 605 the 7 groups are C121 : C5, C605, C11 x (C11 : C5), (C11 x C11) : C5, (C11 x C11) : C5, (C11 x C11) : C5, C55 x C11, where C means Cyclic group and the symbols x and : mean direct and semidirect products respectively. - _Muniru A Asiru_, Nov 11 2017

Muniru A Asiru, Table of n, a(n) for n = 1..198
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.
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), this sequence (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

Warning: The Mma command Select[Range[10^5], FiniteGroupCount[#]==7 &]  gives wrong answers, since FiniteGroupCount[2601] does not return 7. - N. J. A. Sloane, Apr 11 2020

Sequence is { m | A000001(m) = 7 }. - Muniru A Asiru, Nov 11 2017

More terms from Muniru A Asiru, Oct 22 2017

Missing terms added by Muniru A Asiru, Nov 12 2017

A294155 Numbers m such that there are precisely 14 groups of order m.

Original entry on oeis.org

16, 36, 40, 104, 232, 296, 351, 424, 488, 808, 872, 1125, 1192, 1197, 1256, 1384, 1448, 1576, 1755, 1832, 2152, 2216, 2223, 2331, 2344, 2536, 2625, 2792, 2984, 3112, 3176, 3368, 3688, 3861, 4072, 4328, 4329, 4456, 4599, 4875, 4904, 5115, 5187, 5224, 5288, 5301

Offset: 1

Muniru A Asiru, Oct 24 2017

nonn

			For m = 16, the 14 groups of order 16 are C16, C4 x C4, (C4 x C2) : C2, C4 : C4, C8 x C2, C8 : C2, D16, QD16, Q16, C4 x C2 x C2, C2 x D8, C2 x Q8, (C4 x C2) : C2, C2 x C2 x C2 x C2  and for n = 36 the 14 groups of order 36 are C9 : C4, C36, (C2 x C2) : C9, D36, C18 x C2, C3 x (C3 : C4), (C3 x C3) : C4, C12 x C3, (C3 x C3) : C4, S3 x S3, C3 x A4, C6 x S3, C2 x ((C3 x C3) : C2), C6 x C6 where C, D, Q  mean Cyclic group, Dihedral group, Quaternion group of the stated order and S is the Symmetric group of the stated degree. The symbols x and : mean direct and semi-direct products respectively.

Muniru A Asiru, Table of n, a(n) for n = 1..377
H. U. Besche, B. Eick and E. A. O'Brien. The Small Groups Library
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), this sequence (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A294155 := Filtered([1..2015], n -> NumberSmallGroups(n) = 14);

A294156 Numbers m such that there are precisely 15 groups of order m.

Original entry on oeis.org

24, 54, 81, 84, 136, 220, 228, 250, 260, 328, 340, 372, 513, 516, 580, 584, 620, 625, 686, 712, 740, 776, 804, 884, 891, 904, 948, 999, 1060, 1096, 1236, 1375, 1377, 1420, 1460, 1508, 1524, 1544, 1668, 1780, 1812, 1863, 1864, 1911, 1924, 1928, 1940, 1956, 1971, 1972, 2056, 2132, 2180

Offset: 1

Muniru A Asiru, Oct 24 2017

nonn

			For m = 24, the 15 groups of order 24 are C3 : C8, C24, SL(2,3), C3 : Q8, C4 x S3, D24, C2 x (C3 : C4), (C6 x C2) : C2, C12 x C2, C3 x D8, C3 x Q8, S4, C2 x A4, C2 x C2 x S3, C6 x C2 x C2 and for n = 54 the 15 groups of order 54 are D54, C54, C3 x D18, C9 x S3, ((C3 x C3) : C3) : C2, (C9 : C3) : C2, (C9 x C3) : C2, ((C3 x C3) : C3) : C2, C18 x C3, C2 x ((C3 x C3) : C3), C2 x (C9 : C3), C3 x C3 x S3, C3 x ((C3 x C3) : C2), (C3 x C3 x C3) : C2, C6 x C3 x C3 where C, D, Q, S, A and SL mean Cyclic, Dihedral, Quaternion, Symmetric, Alternating and Special Linear group. The symbols x and : mean direct and semi-direct products respectively.

Muniru A Asiru, Table of n, a(n) for n = 1..1061
H. U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library
Gordon Royle, Small Even Order Groups [archived copy]
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), this sequence (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A294156 := Filtered([1..2015], n -> NumberSmallGroups(n) = 15);

Select[ Range@2000, FiniteGroupCount@# == 15 &] (* Robert G. Wilson v, Oct 24 2017 *)

A294156 = { m | A000001(m) = 15 }. - M. F. Hasler, Oct 24 2017

A295161 Numbers m such that there are precisely 16 groups of order m.

Original entry on oeis.org

100, 126, 234, 405, 550, 558, 676, 774, 812, 1098, 1156, 1206, 1218, 1422, 1550, 1746, 1854, 2050, 2502, 2530, 2718, 2826, 2842, 2943, 2982, 3050, 3164, 3364, 3474, 3550, 3798, 3875, 3916, 4014, 4122, 4134, 4214, 4275, 4338, 4401, 4746, 4986, 5094, 5476, 5516, 5566, 5634, 5958, 6066, 6282

Offset: 1

Muniru A Asiru, Nov 15 2017

nonn

			For m = 100, the 16 groups are C25 : C4, C100, C25 : C4, D100, C50 x C2, C5 x (C5 : C4), (C5 x C5) : C4, C20 x C5, C5 x (C5 : C4), (C5 x C5) : C4, (C5 x C5) : C4, (C5 x C5) : C4, D10 x D10, C10 x D10, C2 x ((C5 x C5) : C2), C10 x C10 where C, D mean Cyclic, Dihedral groups of the stated order and the symbols x and : mean direct and semidirect products respectively.

Muniru A Asiru, Table of n, a(n) for n = 1..339
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.
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), this sequence (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

A295161:=Filtered([1..2015],n->NumberSmallGroups(n)=16);

Sequence is { m | A000001(m) = 16 }.

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

A294949 Numbers m such that there are precisely 17 groups of order m.

Original entry on oeis.org

675, 3267, 3549, 9947, 11475, 12625, 14283, 14749, 15525, 17745, 18875, 19575, 22707, 24353, 31725, 35775, 38759, 39039, 39825, 41209, 43561, 45387, 49735

Offset: 1

Muniru A Asiru, Nov 11 2017

nonn

			For m = 675, the 17 groups are C675, C225 x C3, C25 x ((C3 x C3) : C3), C25 x (C9 : C3), (C5 x C5) : C27, C135 x C5, C75 x C3 x C3, C9 x ((C5 x C5) : C3), (C45 x C5) : C3, C3 x ((C5 x C5) : C9), ((C5 x C5) : C9) : C3, (C15 x C15) : C3, C45 x C15, C5 x C5 x ((C3 x C3) : C3), C5 x C5 x (C9 : C3), C3 x C3 x ((C5 x C5) : C3), C15 x C15 x C3 where C means Cyclic group and the symbols x and : mean direct and semidirect products respectively.

Table of n, a(n) for n = 1..23
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.
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), this sequence (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

with(GroupTheory): select(n->NumGroups(n)=17, [$1..150001]); # Muniru A Asiru, Mar 27 2018

Sequence is { m | A000001(m) = 17 }.

More terms from Muniru A Asiru, Nov 17 2017

Incorrect terms removed by Andrew Howroyd, Jan 28 2022

A298909 Numbers m such that there are precisely 18 groups of order m.

Original entry on oeis.org

156, 342, 444, 666, 732, 876, 930, 1164, 1308, 1314, 1830, 1884, 1962, 2172, 2286, 2316, 2748, 2892, 2934, 3258, 3324, 3582, 3675, 3756, 4044, 4125, 4188, 4422, 4476, 4530, 4764, 4878, 4908, 4970, 5050, 5052, 5196, 5430, 5445, 5481, 5484, 5526, 6330, 6492, 6822, 6924

Offset: 1

Muniru A Asiru, Jan 28 2018

nonn

			For m = 156, the 18 groups are (C13 : C4) : C3, C4 x (C13 : C3), C13 x (C3 : C4), C3 x (C13 : C4), C39 : C4, C156, (C13 : C4) : C3, C2 x ((C13 : C3) : C2), C3 x (C13 : C4), C39 : C4, S3 x D26, C2 x C2 x (C13 : C3), C13 x A4, (C26 x C2) : C3, C6 x D26, C26 x S3, D156, C78 x C2 where C, D mean Cyclic, Dihedral groups of the stated order and S, A mean the Symmetric, Alternating groups of the stated degree. The symbols x and : mean direct and semidirect products respectively.

Jorge R. F. F. Lopes, Table of n, a(n) for n = 1..283
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.
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), this sequence (k=18), A298910 (k=19), A298911 (k=20).

Filtered([1..2015], n -> NumberSmallGroups(n) = 18);

with(GroupTheory):
for n from 1 to 10^4 do if NumGroups(n) = 18 then print(n); fi; od;

Sequence is { m | A000001(m) = 18 }.

cp's OEIS Frontend

A249555 Numbers m such that there are precisely 12 groups of order m.

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A292896 Numbers m such that there are precisely 13 groups of order m.

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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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Magma

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A249550 Numbers m such that there are precisely 7 groups of order m.

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A294155 Numbers m such that there are precisely 14 groups of order m.

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GAP

A294156 Numbers m such that there are precisely 15 groups of order m.

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GAP

Mathematica

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A295161 Numbers m such that there are precisely 16 groups of order m.

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Programs

GAP

Formula

A000019 Number of primitive permutation groups of degree n.

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