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a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 17.
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%I #9 Nov 09 2022 07:55:11

%S 131071,8589737985,2821088318560,281468534292480,4999961852994576,

%T 184880022956829600,725978907114673600,9223160931695984640,

%U 40479533921803813920,327672500035999538160,602267704294826658336,6058148592249392332800,7268068365187380400720

%N a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 17.

%H Amiram Eldar, <a href="/A161215/b161215.txt">Table of n, a(n) for n = 1..10000</a>

%H Jin Ho Kwak and Jaeun Lee, <a href="https://doi.org/10.1142/9789812799890_0005">Enumeration of graph coverings, surface branched coverings and related group theory</a>, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.

%F From _Amiram Eldar_, Nov 08 2022: (Start)

%F a(n) = 131071 * A161167(n).

%F Sum_{k=1..n} a(k) ~ c * n^16, where c = (131071/16) * Product_{p prime} (1 + (p^15-1)/((p-1)*p^16)) = 15921.65841... .

%F Sum_{k>=1} 1/a(k) = (zeta(15)*zeta(16)/131071) * Product_{p prime} (1 - 2/p^16 + 1/p^31) = 7.6295695155...*10^(-6). (End)

%t f[p_, e_] := p^(15*e - 15) * (p^16-1) / (p-1); a[1] = 131071; a[n_] := 131071 * Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Nov 08 2022 *)

%o (PARI) a(n) = {my(f = factor(n)); 131071 * prod(i = 1, #f~, (f[i,1]^16 - 1)*f[i,1]^(15*f[i,2] - 15)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022

%Y Cf. A000010, A013673, A013674, A161167.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Nov 19 2009