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%I #71 May 13 2023 23:50:17
%S 8,12,18,20,27,50,52,68,98,116,125,135,148,164,171,212,242,244,273,
%T 292,297,333,338,343,356,388,399,404,436,452,459,548,578,596,621,628,
%U 651,657,692,722,724,741,772,777,783,788,825,855,875,916,932,964,981
%N Numbers m such that there are precisely 5 groups of order m.
%C For m = 2*p^2 (p prime), there are precisely 5 groups of order m, so A079704 and A143928 (p odd prime) are two subsequences. - _Bernard Schott_, Dec 10 2021
%C For m = p^3, p prime, there are also 5 groups of order m, so A030078, where these groups are described, is another subsequence. - _Bernard Schott_, Dec 11 2021
%H Jorge R. F. F. Lopes, <a href="/A054397/b054397.txt">Table of n, a(n) for n = 1..2035</a>, (terms 1..120 from Muniru A Asiru and Georg Fischer).
%H H. U. Besche, B. Eick and E. A. O'Brien, <a href="http://www.icm.tu-bs.de/ag_algebra/software/small/">The Small Groups Library</a>
%H Gordon Royle, <a href="https://web.archive.org/web/20171109093930/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 { k | A000001(k) = 5 }. - _Muniru A Asiru_, Nov 03 2017
%e For m = 8, the 5 groups of order 8 are C8, C4 x C2, D8, Q8, C2 x C2 x C2 and for m = 12 the 5 groups of order 12 are C3 : C4, C12, A4, D12, C6 x C2 where C, D, Q mean cyclic, dihedral, quaternion groups of the stated order and A is the alternating group of the stated degree. The symbols x and : mean direct and semidirect products respectively. - _Muniru A Asiru_, Nov 03 2017
%t Select[Range[10^4], FiniteGroupCount[#] == 5 &] (* _Robert Price_, May 23 2019 *)
%o (GAP) A054397 := Filtered([1..2015], n -> NumberSmallGroups(n) = 5); # _Muniru A Asiru_, Nov 03 2017
%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), this sequence (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), A298909 (k=18), A298910 (k=19), A298911 (k=20).
%K nonn
%O 1,1
%A _N. J. A. Sloane_, May 21 2000
%E More terms from _Christian G. Bower_, May 25 2000
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