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A160908 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 9.

%I A160908 #41 Apr 01 2025 19:46:25
%S A160908 1,255,3280,32640,97656,836400,960800,4177920,7173360,24902280,
%T A160908 21435888,107059200,67977560,245004000,320311680,534773760,435984840,
%U A160908 1829206800,943531280,3187491840,3151424000,5466151440,3559590240,13703577600,7629375000,17334277800
%N A160908 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 9.
%C A160908 a(n) is the number of lattices L in Z^8 such that the quotient group Z^8 / L is C_n. - _Álvar Ibeas_, Oct 30 2015
%H A160908 G. C. Greubel, <a href="/A160908/b160908.txt">Table of n, a(n) for n = 1..5000</a>
%H A160908 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.
%H A160908 <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F A160908 a(n) = J_8(n)/J_1(n) = J_8(n)/phi(n) = A069093(n)/A000010(n), where J_k is the k-th Jordan totient function. - _Enrique Pérez Herrero_, Oct 28 2010
%F A160908 From _Álvar Ibeas_, Oct 30 2015: (Start)
%F A160908 Multiplicative with a(p^e) = p^(7e-7) * (p^8-1) / (p-1).
%F A160908 For squarefree n, a(n) = A000203(n^7). (End)
%F A160908 From _Amiram Eldar_, Nov 08 2022: (Start)
%F A160908 Sum_{k=1..n} a(k) ~ c * n^8, where c = (1/8) * Product_{p prime} (1 + (p^7-1)/((p-1)*p^8)) = 0.2423008904... .
%F A160908 Sum_{k>=1} 1/a(k) = zeta(7)*zeta(8) * Product_{p prime} (1 - 2/p^8 + 1/p^15) = 1.004270064601... . (End)
%F A160908 a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^8). - _Ridouane Oudra_, Apr 01 2025
%t A160908 A160908[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(9-1)/EulerPhi[n]&] (* _Enrique Pérez Herrero_, Oct 28 2010 *)
%t A160908 f[p_, e_] := p^(7*e - 7) * (p^8-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* _Amiram Eldar_, Nov 08 2022 *)
%o A160908 (PARI) vector(30, n, sumdiv(n^7, d, if(ispower(d, 8), moebius(sqrtnint(d, 8))*sigma(n^7/d), 0))) \\ _Altug Alkan_, Oct 30 2015
%o A160908 (PARI) a(n) = {f = factor(n); for (i=1, #f~, p = f[i,1]; f[i,1] = p^(7*f[i,2]-7)*(p^8-1)/(p-1); f[i,2] = 1;); factorback(f);} \\ _Michel Marcus_, Nov 12 2015
%Y A160908 Column 8 of A263950.
%Y A160908 Cf. A000010, A000203, A008683, A013665, A013666, A069093.
%K A160908 nonn,mult
%O A160908 1,2
%A A160908 _N. J. A. Sloane_, Nov 19 2009
%E A160908 Definition corrected by _Enrique Pérez Herrero_, Oct 28 2010