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Numbers m such that there are precisely 7 groups of order m.
20

%I #40 May 13 2023 23:50:26

%S 375,605,903,1705,2255,2601,2667,3081,3355,3905,3993,4235,4431,4515,

%T 4805,5555,6123,6355,6375,6765,7077,7205,7865,7917,7959,8305,8405,

%U 8625,8841,9455,9723,9933,9955,10285,10505,10875,11005,11487,11495,11571,11605,11715,11935,12207,12505,13005,13053,13251,13255,13335,13805,14133

%N Numbers m such that there are precisely 7 groups of order m.

%H Muniru A Asiru, <a href="/A249550/b249550.txt">Table of n, a(n) for n = 1..198</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) = 7 }. - _Muniru A Asiru_, Nov 11 2017

%e For m = 375, the 7 groups are C375, ((C5 x C5) : C5) : C3, C75 x C5, C3 x ((C5 x C5) : C5), C3 x (C25 : C5), C5 x ((C5 x C5) : C3), C15 x C5 x C5 and for n = 605 the 7 groups are C121 : C5, C605, C11 x (C11 : C5), (C11 x C11) : C5, (C11 x C11) : C5, (C11 x C11) : C5, C55 x C11, where C means Cyclic group and the symbols x and : mean direct and semidirect products respectively. - _Muniru A Asiru_, Nov 11 2017

%t Warning: The Mma command Select[Range[10^5], FiniteGroupCount[#]==7 &] gives wrong answers, since FiniteGroupCount[2601] does not return 7. - _N. J. A. Sloane_, Apr 11 2020

%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), this sequence (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_, Nov 01 2014

%E More terms from _Muniru A Asiru_, Oct 22 2017

%E Missing terms added by _Muniru A Asiru_, Nov 12 2017