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A104488 Number of Hamiltonian groups of order n. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,72

REFERENCES

R. D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.

J. C. Lennox and S. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

B. Horvat, G. Jaklic and T. Pisanski, On the number of Hamiltonian groups, arXiv:math/0503183 [math.CO], 2005.

T. Pisanski and T.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.

a(8*n) = A000688(A000265(n)), a(n) = 0 for n mod 8 <> 0. - Andrew Howroyd, Aug 08 2018

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-Fran├žois 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

Cf. A000265, A000688, A104404, A104404, A104452, A104453.

Sequence in context: A245515 A327170 A024362 * A244413 A318655 A056626

Adjacent sequences:  A104485 A104486 A104487 * A104489 A104490 A104491

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

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Last modified January 19 12:56 EST 2020. Contains 331049 sequences. (Running on oeis4.)