cp's OEIS Frontend

A104407 Number of Hamiltonian groups of order <= n.

%I A104407 #21 Feb 16 2025 08:32:56
%S A104407 0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,
%T A104407 4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,
%U A104407 8,8,8,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,12,12,12,12,12
%N A104407 Number of Hamiltonian groups of order <= n.
%D A104407 Robert D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.
%D A104407 John C. Lennox and Stewart. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.
%H A104407 Amiram Eldar, <a href="/A104407/b104407.txt">Table of n, a(n) for n = 1..10000</a>
%H A104407 Boris Horvat, Gašper Jaklič, and Tomaž Pisanski, <a href="https://hrcak.srce.hr/clanak/1339">On the number of hamiltonian groups</a>, Mathematical Communications, Vol. 10, No. 1 (2005), pp. 89-94; <a href="https://arxiv.org/abs/math/0503183">arXiv preprint</a>, arXiv:math/0503183 [math.CO], 2005.
%H A104407 Tomaž Pisanski and Thomas W. Tucker, <a href="https://doi.org/10.1016/0012-365X(89)90173-8">The genus of low rank hamiltonian groups</a>, Discrete Math. 78 (1989), 157-167.
%H A104407 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianGroup.html">Hamiltonian Group</a>.
%F A104407 a(n) ~ c * n, where c = A021002 * A048651 / 4 = 0.16568181590156732257... . - _Amiram Eldar_, Oct 03 2023
%t A104407 orders[n_]:=Map[Last, FactorInteger[n]]; a[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/;Mod[n, 8]==0:=a[e[n]]; h[n_]:=0; numberOfHamiltonianGroupsOfOrderLEQThanN[n_]:=Map[Apply[Plus, # ]&, Table[Take[Map[h, Table[i, {i, 1, n}]], i], {i, 1, n}]];
%Y A104407 Partial sums of A104488.
%Y A104407 Cf. A000688, A021002, A048651, A063966, A104404, A104452, A104453.
%K A104407 nonn,easy
%O A104407 1,16
%A A104407 Boris Horvat (Boris.Horvat(AT)fmf.uni-lj.si), Gasper Jaklic (Gasper.Jaklic(AT)fmf.uni-lj.si), _Tomaz Pisanski_, Apr 19 2005