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%I #27 Sep 23 2023 03:48:09
%S 1,1,1,2,1,1,1,4,2,1,1,2,1,1,1,6,1,2,1,2,1,1,1,4,2,1,3,2,1,1,1,8,1,1,
%T 1,4,1,1,1,4,1,1,1,2,2,1,1,6,2,2,1,2,1,3,1,4,1,1,1,2,1,1,2,12,1,1,1,2,
%U 1,1,1,8,1,1,2,2,1,1,1,6,5,1,1,2,1,1,1,4,1,2,1,2,1,1,1,8,1,2,2,4,1,1
%N Number of groups of order n all of whose subgroups are normal.
%C A finite non-Abelian group has all of its subgroups normal precisely when it is the direct product of the quaternion group of order 8, a (possibly trivial) elementary Abelian 2-group, and an Abelian group of odd order. [Carmichael, p. 114] - _Eric M. Schmidt_, Jan 12 2014
%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 Hans Havermann, <a href="/A104404/b104404.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 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 The number a(n) of all groups of order n all of whose subgroups are normal is given as a(n) = b(n) + h(n), where b(n) denotes the number of Abelian groups of order n and h(n) denotes the number of Hamiltonian groups of order n.
%F a(n) = A000688(n) + A104488(n). - _Andrew Howroyd_, Aug 08 2018
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A021002 * (1 + A048651/4) = 2.46053840757488111675... . - _Amiram Eldar_, Sep 23 2023
%t orders[n_]:=Map[Last, FactorInteger[n]]; b[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/;Mod[n, 8]==0:=b[e[n]]; h[n_]:=0; a[n_]:= b[n]+h[n];
%o (PARI) a(n)={my(e=valuation(n, 2)); my(f=factor(n/2^e)[, 2]); prod(i=1, #f, numbpart(f[i]))*(numbpart(e) + (e>=3))} \\ _Andrew Howroyd_, Aug 08 2018
%Y Cf. A000001, A000688, A021002, A048651, A104488.
%K nonn,easy,mult
%O 1,4
%A Boris Horvat (Boris.Horvat(AT)fmf.uni-lj.si), Gasper Jaklic (Gasper.Jaklic(AT)fmf.uni-lj.si), _Tomaz Pisanski_, Apr 19 2005
%E Keyword:mult added by _Andrew Howroyd_, Aug 08 2018
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