A104488 Number of Hamiltonian groups of order n.
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
Offset: 1
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.
Programs
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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 *)
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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
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