A298910 - OEIS

%I #16 May 13 2023 23:51:25

%S 1029,5145,6591,7803,8001,11319,11739,12789,17157,17493,20577,21567,

%T 23667,23877,27993,31311,32955,33411,34671,34713,39015,39753,40005,

%U 42189,42861,45675,47691,48363,49833

%N Numbers m such that there are precisely 19 groups of order m.

%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) = 19 }.

%e For m = 1029, the 19 groups are C1029, C147 x C7, C3 x ((C7 x C7) : C7), C3 x (C49 : C7), C21 x C7 x C7, C343 : C3, C49 x (C7 : C3), C7 x (C49 : C3), (C49 x C7) : C3, (C49 x C7) : C3, ((C7 x C7) : C7) : C3, ((C7 x C7) : C7) : C3, ((C7 x C7) : C7) : C3, (C49 : C7) : C3, C7 x ((C7 x C7) : C3), C7 x ((C7 x C7) : C3), (C7 x C7 x C7) : C3, (C7 x C7 x C7) : C3, C7 x C7 x (C7 : C3) where C means the Cyclic group of the stated order and the symbols x and : mean direct and semidirect products respectively.

%p with(GroupTheory):

%p for n from 1 to 3*10^5 do if NumGroups(n) = 19 then print(n); fi; od;

%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), A298909 (k=18), this sequence (k=19), A298911 (k=20).

%K nonn

%O 1,1

%A _Muniru A Asiru_, Jan 28 2018

%E Shortened to remove possibly incorrect terms by _Andrew Howroyd_, Jan 28 2022