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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-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 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.)