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A056291 Number of primitive (period n) n-bead necklaces with exactly six different colored beads.

%I A056291 #17 Jun 06 2018 14:31:47
%S A056291 0,0,0,0,0,120,2160,23940,211680,1643544,11748240,79419060,516257280,
%T A056291 3262440960,20193277104,123071683140,741419995680,4427489935680,
%U A056291 26264144909520,155018839412052,911509010152560,5344538372696880,31272099902089200,182707081042818360
%N A056291 Number of primitive (period n) n-bead necklaces with exactly six different colored beads.
%C A056291 Turning over the necklace is not allowed.
%D A056291 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
%H A056291 Alois P. Heinz, <a href="/A056291/b056291.txt">Table of n, a(n) for n = 1..1000</a>
%F A056291 Sum mu(d)*A056286(n/d) where d|n.
%p A056291 with(numtheory):
%p A056291 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A056291       add(mobius(n/d)*k^d, d=divisors(n))/n)
%p A056291     end:
%p A056291 a:= n-> add(b(n, 6-j)*binomial(6, j)*(-1)^j, j=0..6):
%p A056291 seq(a(n), n=1..30);  # _Alois P. Heinz_, Jan 25 2015
%t A056291 b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^# &]/n];
%t A056291 a[n_] := Sum[b[n, 6 - j]*Binomial[6, j]*(-1)^j, {j, 0, 6}];
%t A056291 Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Jun 06 2018, after _Alois P. Heinz_ *)
%Y A056291 Cf. A032164.
%Y A056291 Column k=6 of A254040.
%K A056291 nonn
%O A056291 1,6
%A A056291 _Marks R. Nester_