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

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

Author

Muniru A Asiru, Jan 28 2018

Keywords

• nonn

Examples

			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.
		

Links

• 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

Crossrefs

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).

Programs

GAP

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

Maple

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

Formula

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