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%I #18 May 14 2023 02:32:22
%S 156,342,444,666,732,876,930,1164,1308,1314,1830,1884,1962,2172,2286,
%T 2316,2748,2892,2934,3258,3324,3582,3675,3756,4044,4125,4188,4422,
%U 4476,4530,4764,4878,4908,4970,5050,5052,5196,5430,5445,5481,5484,5526,6330,6492,6822,6924
%N Numbers m such that there are precisely 18 groups of order m.
%H Jorge R. F. F. Lopes, <a href="/A298909/b298909.txt">Table of n, a(n) for n = 1..283</a>
%H H. U. Besche, B. Eick and E. A. O'Brien, <a href="http://dx.doi.org/10.1142/S0218196702001115">A Millennium Project: Constructing Small Groups</a>, Internat. J. Algebra and Computation, 12 (2002), 623-644.
%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/cubcay/">Numbers of Small Groups</a>
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F Sequence is { m | A000001(m) = 18 }.
%e 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.
%p with(GroupTheory):
%p for n from 1 to 10^4 do if NumGroups(n) = 18 then print(n); fi; od;
%o (GAP) Filtered([1..2015], n -> NumberSmallGroups(n) = 18);
%Y 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).
%K nonn
%O 1,1
%A _Muniru A Asiru_, Jan 28 2018
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