%I #15 Nov 09 2022 07:55:07

%S 7,21,28,42,42,84,56,84,84,126,84,168,98,168,168,168,126,252,140,252,

%T 224,252,168,336,210,294,252,336,210,504,224,336,336,378,336,504,266,

%U 420,392,504,294,672,308,504,504,504,336,672,392,630,504,588,378,756

%N a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 3.

%H Antti Karttunen, <a href="/A160890/b160890.txt">Table of n, a(n) for n = 1..16384</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) = 7 * A001615(n).

%F Sum_{k=1..n} a(k) ~ c * n^2 + O(n*log(n)), where c = 105/(2*Pi^2) = 5.319362... . (End)

%t With[{b = 3}, Table[((2^b - 1)/EulerPhi[n]) DivisorSum[n, MoebiusMu[n/#] #^(b - 1) &], {n, 54}]] (* _Michael De Vlieger_, Nov 23 2017 *)

%t f[p_, e_] := (p + 1)*p^(e - 1); a[1] = 7; a[n_] := 7*Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 08 2022 *)

%o (PARI) A160890(n) = ((7/eulerphi(n))*sumdiv(n,d,moebius(n/d)*(d^2))); \\ _Antti Karttunen_, Nov 23 2017

%o (PARI) a(n) = {my(f = factor(n)); 7 * prod(i = 1, #f~, (f[i,1] + 1)*f[i,1]^(f[i,2] - 1));} \\ _Amiram Eldar_, Nov 08 2022

%Y Cf. A000010, A001615.

%K nonn

%O 1,1

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