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A104488
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Number of Hamiltonian groups of order n.
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5
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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
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OFFSET
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1,72
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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MATHEMATICA
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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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PROG
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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
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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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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STATUS
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approved
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