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%I M3860 N1582 #71 May 07 2021 02:13:42
%S 1,5,15,45,165,629,2635,11165,48915,217045,976887,4438925,20346485,
%T 93900245,435970995,2034505661,9536767665,44878791365,211927736135,
%U 1003867701485,4768372070757,22706531350485,108372083629275,518301258916445
%N Number of n-bead necklaces with 5 colors.
%C From _Richard L. Ollerton_, May 07 2021: (Start)
%C Here, as in A000031, turning over is not allowed.
%C (1/n) * Dirichlet convolution of phi(n) and 5^n, n>0. (End)
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 162.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a).
%H T. D. Noe, <a href="/A001869/b001869.txt">Table of n, a(n) for n=0..200</a>
%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=5">Encyclopedia of Combinatorial Structures 5</a>
%H Juhani Karhumäki, S. Puzynina, M. Rao, and M. A. Whiteland, <a href="https://arxiv.org/abs/1605.03319">On cardinalities of k-abelian equivalence classes</a>, arXiv preprint arXiv:1605.03319 [math.CO], 2016.
%H J. Riordan, <a href="/A001867/a001867.pdf">Letter to N. J. A. Sloane, Jul. 1978</a>
%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)*5^(n/d), n > 0.
%F G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 5*x^n)/n. - _Herbert Kociemba_, Nov 01 2016
%F a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 5^gcd(n,k). - _Ilya Gutkovskiy_, Apr 17 2021
%F a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} 5^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - _Richard L. Ollerton_, May 07 2021
%t CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-5*x^i]/i,{i,1,mx}],{x,0,mx}],x] (* _Herbert Kociemba_, Nov 01 2016 *)
%t k=5; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/n, {n, 1, 30}], 1] (* _Robert A. Russell_, Sep 21 2018 *)
%o (PARI) a(n) = if (n, sumdiv(n, d, eulerphi(d)*5^(n/d))/n, 1); \\ _Michel Marcus_, Nov 01 2016
%Y Column 5 of A075195.
%Y Cf. A054612.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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