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%I #24 Nov 08 2022 08:07:50
%S 1,4095,265720,8386560,61035156,1088123400,2306881200,17175674880,
%T 47071500840,249938963820,313842837672,2228476723200,1941507093540,
%U 9446678514000,16218261652320,35175782154240,36413889826860
%N a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 13.
%C a(n) is the number of lattices L in Z^12 such that the quotient group Z^12 / L is C_n. - _Álvar Ibeas_, Nov 26 2015
%H Álvar Ibeas, <a href="/A160972/b160972.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.
%H <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F a(n) = J_12(n)/J_1(n) where J_12 and J_1(n) = A000010(n) are Jordan functions. - _R. J. Mathar_, Jul 12 2011
%F From _Álvar Ibeas_, Nov 26 2015: (Start)
%F Multiplicative with a(p^e) = p^(11e-11) * (p^12-1) / (p-1).
%F For squarefree n, a(n) = A000203(n^11). (End)
%F From _Amiram Eldar_, Nov 08 2022: (Start)
%F Sum_{k=1..n} a(k) ~ c * n^12, where c = (1/12) * Product_{p prime} (1 + (p^11-1)/((p-1)*p^12)) = 0.1619398772... .
%F Sum_{k>=1} 1/a(k) = zeta(11)*zeta(12) * Product_{p prime} (1 - 2/p^12 + 1/p^23) = 1.0002481006668... . (End)
%t b = 13; Table[Sum[MoebiusMu[n/d] d^(b - 1)/EulerPhi@ n, {d, Divisors@ n}], {n, 17}] (* _Michael De Vlieger_, Nov 27 2015 *)
%t f[p_, e_] := p^(11*e - 11) * (p^12-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Nov 08 2022 *)
%o (PARI) vector(100, n, sumdiv(n^11, d, if(ispower(d, 12), moebius(sqrtnint(d, 12))*sigma(n^11/d), 0))) \\ _Altug Alkan_, Nov 26 2015
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^12 - 1)*f[i,1]^(11*f[i,2] - 11)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022
%Y Column 12 of A263950.
%Y Cf. A000010, A000203, A013669, A013670.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_, Nov 19 2009
%E Definition corrected by _Enrique Pérez Herrero_, Oct 30 2010
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