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a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 9.
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%I #18 Nov 09 2022 07:55:45

%S 511,130305,1676080,16679040,49902216,427400400,490968800,2134917120,

%T 3665586960,12725065080,10953738768,54707251200,34736533160,

%U 125197044000,163679268480,273269391360,222788253240,934724674800,482144484080,1628808330240,1610377664000

%N a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 9.

%H Amiram Eldar, <a href="/A160956/b160956.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from G. C. Greubel)

%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) = 511*A160908(n). - _R. J. Mathar_, Mar 16 2016

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

%F Sum_{k=1..n} a(k) ~ c * n^8, where c = (511/8) * Product_{p prime} (1 + (p^7-1)/((p-1)*p^8)) = 123.8157549... .

%F Sum_{k>=1} 1/a(k) = (zeta(7)*zeta(8)/511) * Product_{p prime} (1 - 2/p^8 + 1/p^15) = 0.001965303453... . (End)

%t A160956[n_] := DivisorSum[n, MoebiusMu[n/#]*#^(9 - 1)/EulerPhi[n] &]; Table[511*A160956[n], {n, 1, 50}] (* _G. C. Greubel_, Dec 12 2017 *)

%t f[p_, e_] := p^(7*e - 7) * (p^8-1) / (p-1); a[1] = 511; a[n_] := 511 * Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Nov 08 2022 *)

%o (PARI) a(n) = {my(f = factor(n)); 511 * prod(i = 1, #f~, (f[i,1]^8 - 1)*f[i,1]^(7*f[i,2] - 7)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022

%Y Cf. A000010, A013665, A013666, A160908.

%K nonn

%O 1,1

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