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A054627 Number of n-bead necklaces with 8 colors.

%I A054627 #40 Feb 16 2025 08:32:42
%S A054627 1,8,36,176,1044,6560,43800,299600,2097684,14913200,107377488,
%T A054627 780903152,5726645688,42288908768,314146329192,2345624810432,
%U A054627 17592187093524,132458812569728,1000799924679192,7585009898729264,57646075284033552,439208192231379680
%N A054627 Number of n-bead necklaces with 8 colors.
%H A054627 Alois P. Heinz, <a href="/A054627/b054627.txt">Table of n, a(n) for n = 0..1000</a>
%H A054627 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Necklace.html">Necklace.</a>
%H A054627 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%F A054627 a(n) = (1/n)*Sum_{d|n} phi(d)*8^(n/d), n > 0.
%F A054627 G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 8*x^n)/n. - _Herbert Kociemba_, Nov 02 2016
%F A054627 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 8^gcd(n,k). - _Ilya Gutkovskiy_, Apr 17 2021
%e A054627 G.f. = 1 + 8*x + 36*x^2 + 176*x^3 + 1044*x^4 + 6560*x^5 + 43800*x^6 + ...
%p A054627 with(combstruct):A:=[N,{N=Cycle(Union(Z$8))},unlabeled]: seq(count(A,size=n),n=0..20); # _Zerinvary Lajos_, Dec 05 2007
%t A054627 mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-8*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* _Herbert Kociemba_, Nov 02 2016 *)
%t A054627 k=8; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/n, {n, 1, 30}], 1] (* _Robert A. Russell_, Sep 21 2018 *)
%o A054627 (PARI) a(n)=if(n==0, 1, 1/n*sumdiv(n, d, eulerphi(d)*8^(n/d))); \\ _Altug Alkan_, Sep 21 2018
%Y A054627 Column 8 of A075195.
%Y A054627 Column k=1 of A184294.
%Y A054627 Cf. A054615.
%K A054627 nonn
%O A054627 0,2
%A A054627 _N. J. A. Sloane_, Apr 16 2000
%E A054627 Edited by _Christian G. Bower_, Sep 07 2002
%E A054627 a(0) corrected by _Herbert Kociemba_, Nov 02 2016