A160898 - OEIS

%I #11 Nov 09 2022 07:56:50

%S 127,8001,46228,256032,496062,2912364,2490216,8193024,11233404,

%T 31251906,22498812,93195648,51083718,156883608,180566568,262176768,

%U 191591946,707704452,331934820,1000060992,906438624,1417425156,854570808,2982260736,1550193750,3218274234

%N a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 7.

%H Amiram Eldar, <a href="/A160898/b160898.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 a(n) = 127*A160895(n). - _R. J. Mathar_, Mar 15 2016

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

%F Sum_{k=1..n} a(k) ~ c * n^6, where c = (127/6) * Product_{p prime} (1 + (p^5-1)/((p-1)*p^6)) = 40.6863089361... .

%F Sum_{k>=1} 1/a(k) = (zeta(5)*zeta(6)/127) * Product_{p prime} (1 - 2/p^6 + 1/p^11) = 0.008027649545... . (End)

%t f[p_, e_] := p^(5*e - 5) * (p^6-1) / (p-1); a[1] = 127; a[n_] := 127 * Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* _Amiram Eldar_, Nov 08 2022 *)

%o (PARI) a(n) = {my(f = factor(n)); 127 * prod(i = 1, #f~, (f[i,1]^6 - 1)*f[i,1]^(5*f[i,2] - 5)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022

%Y Cf. A000010, A013663, A013664, A160895.

%K nonn

%O 1,1

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