A294156 - OEIS
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
EXAMPLE
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.
MATHEMATICA
Select[ Range@2000, FiniteGroupCount@# == 15 &] (* Robert G. Wilson v, Oct 24 2017 *)
PROG
(GAP) A294156 := Filtered([1..2015], n -> NumberSmallGroups(n) = 15);
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), this sequence (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).