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%I #15 Oct 03 2023 13:11:59
%S 1,2,3,5,6,7,8,12,14,15,16,18,19,20,21,27,28,30,31,33,34,35,36,40,42,
%T 43,46,48,49,50,51,59,60,61,62,66,67,68,69,73,74,75,76,78,80,81,82,88,
%U 90,92,93,95,96,99,100,104,105,106,107,109,110,111,113,125,126,127
%N Number of groups of order <= n all of whose subgroups are normal.
%D Robert D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.
%D John C. Lennox and Stewart. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.
%H Amiram Eldar, <a href="/A104452/b104452.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbelianGroup.html">Abelian Group</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianGroup.html">Hamiltonian Group</a>.
%F a(n) ~ c * n, where c = A021002 * (1 + A048651/4) = 2.46053840757488111675... . - _Amiram Eldar_, Oct 03 2023
%t 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; numberOfAbelianGroupsOfOrderLEQThanN[n_]:=Map[Apply[Plus, # ]&, Table[Take[Map[a, Table[i, {i, 1, n}]], i], {i, 1, n}]]; numberOfHamiltonianGroupsOfOrderLEQThanN[n_]:=Map[Apply[Plus, # ]&, Table[Take[Map[h, Table[i, {i, 1, n}]], i], {i, 1, n}]]; numberOfAllGroupsOfOrderLEQThanN[n_]:=numberOfAbelianGroupsOfOrderLEQThanN[n] +numberOfHamiltonianGroupsOfOrderLEQThanN[n];
%Y Partial sums of A104404.
%Y Cf. A000688, A021002, A048651, A063966, A104488, A104407, A104453.
%K nonn,easy
%O 1,2
%A Boris Horvat (Boris.Horvat(AT)fmf.uni-lj.si), Gasper Jaklic (Gasper.Jaklic(AT)fmf.uni-lj.si), _Tomaz Pisanski_, Apr 19 2005
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