A161004 - OEIS

%I #8 Nov 09 2022 07:55:31

%S 4095,8382465,362706435,8583644160,49987791945,742460072445,

%T 1349525501415,8789651619840,21417452280315,102325010111415,

%U 116835129114795,760279114183680,611574734464785,2762478701396505,4427568695944485,9000603258716160,8771463461234565

%N a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 12.

%H Amiram Eldar, <a href="/A161004/b161004.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) = 4095 * A160960(n).

%F Sum_{k=1..n} a(k) ~ c * n^11, where c = (4095/11) * Product_{p prime} (1 + (p^10-1)/((p-1)*p^11)) =  723.3106628... .

%F Sum_{k>=1} 1/a(k) = (zeta(10)*zeta(11)/4095) * Product_{p prime} (1 - 2/p^11 + 1/p^21) = 0.0002443224366... . (End)

%t f[p_, e_] := p^(10*e - 10) * (p^11-1) / (p-1); a[1] = 4095; a[n_] := 4095 * Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* _Amiram Eldar_, Nov 08 2022 *)

%o (PARI) a(n) = {my(f = factor(n)); 4095 * prod(i = 1, #f~, (f[i,1]^11 - 1)*f[i,1]^(10*f[i,2] - 10)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022

%Y Cf. A000010, A013668, A013669, A160960.

%K nonn

%O 1,1

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