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A054629 Number of n-bead necklaces with 10 colors. 2

%I

%S 1,10,55,340,2530,20008,166870,1428580,12501280,111111340,1000010044,

%T 9090909100,83333418520,769230769240,7142857857190,66666666680272,

%U 625000006251280,5882352941176480,55555555611222370,526315789473684220

%N Number of n-bead necklaces with 10 colors.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Necklace.html">Necklace.</a>

%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>

%F a(n) = (1/n)*Sum_{d|n} phi(d)*10^(n/d) = A054617(n)/n, n > 0.

%F G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 10*x^n)/n. - _Herbert Kociemba_, Nov 02 2016

%e G.f. = 1 + 10*x + 55*x^2 + 340*x^3 + 2530*x^4 + 20008*x^5 + 166870*x^6 + ...

%p with(combstruct):A:=[N,{N=Cycle(Union(Z$10))},unlabeled]: seq(count(A,size=n),n=0..19); # _Zerinvary Lajos_, Dec 05 2007

%t mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-10*x^i]/i, {i, 1, mx}], {x, 0, mx}], x] (* _Herbert Kociemba_, Nov 02 2016 *)

%t k=10; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/n, {n, 1, 30}], 1] (* _Robert A. Russell_, Sep 21 2018 *)

%o (PARI) a(n)=if(n==0, 1, 1/n*sumdiv(n, d, eulerphi(d)*10^(n/d))); \\ _Altug Alkan_, Sep 21 2018

%Y Column 10 of A075195.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Apr 16 2000

%E Edited by _Christian G. Bower_, Sep 07 2002

%E a(0) corrected by _Herbert Kociemba_, Nov 02 2016

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Last modified October 20 10:32 EDT 2019. Contains 328257 sequences. (Running on oeis4.)