A054397 - OEIS

A054397

Numbers m such that there are precisely 5 groups of order m.

8, 12, 18, 20, 27, 50, 52, 68, 98, 116, 125, 135, 148, 164, 171, 212, 242, 244, 273, 292, 297, 333, 338, 343, 356, 388, 399, 404, 436, 452, 459, 548, 578, 596, 621, 628, 651, 657, 692, 722, 724, 741, 772, 777, 783, 788, 825, 855, 875, 916, 932, 964, 981

COMMENTS

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

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

For m squarefree, there are 5 groups of order m if and only if all of the following hold: 3|m, there are exactly two prime factors p,q of m such that p,q = 1 mod 3, no other relations of the form p' = 1 mod q' hold for p',q' prime factors of m. - Robin Jones, May 27 2025

EXAMPLE

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

MATHEMATICA

Select[Range[10^4], FiniteGroupCount[#] == 5 &] (* Robert Price, May 23 2019 *)

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), 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).

Cf. A384370 (squarefree numbers in this sequence).