OFFSET
1,72
REFERENCES
Robert D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.
John C. Lennox and Stewart. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
Boris Horvat, Gašper Jaklič, and Tomaž Pisanski, On the number of hamiltonian groups, Mathematical Communications, Vol. 10, No. 1 (2005), pp. 89-94; arXiv preprint, arXiv:math/0503183 [math.CO], 2005.
Tomaž Pisanski and Thomas W. Tucker, The genus of low rank hamiltonian groups, Discrete Math. 78 (1989), 157-167.
Eric Weisstein's World of Mathematics, Hamiltonian Group.
FORMULA
Let n = 2^e*o, where e = e(n) >= 0 and o = o(n) is an odd number. The number h(n) of Hamiltonian groups of order n is given by h(n) = 0, if e(n) < 3 and h(n) = a(o(n)), otherwise, where a(n) = A000688(n) denotes the number of Abelian groups of order n.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A021002 * A048651 / 4 = 0.16568181590156732257... . - Amiram Eldar, Sep 23 2023
MATHEMATICA
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;
(* Second program: *)
a[n_] := If[Mod[n, 8]==0, FiniteAbelianGroupCount[n/2^IntegerExponent[n, 2]], 0]; Array[a, 102] (* Jean-François Alcover, Sep 14 2019 *)
PROG
(PARI) a(n)={my(e=valuation(n, 2)); if(e<3, 0, my(f=factor(n/2^e)[, 2]); prod(i=1, #f, numbpart(f[i])))} \\ Andrew Howroyd, Aug 08 2018
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Boris Horvat (Boris.Horvat(AT)fmf.uni-lj.si), Gasper Jaklic (Gasper.Jaklic(AT)fmf.uni-lj.si), Tomaz Pisanski, Apr 19 2005
STATUS
approved