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A161010 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 14.

%I A161010 #35 Apr 02 2025 15:16:44
%S A161010 1,8191,797161,33550336,305175781,6529545751,16148168401,137422176256,
%T A161010 423644039001,2499694822171,3452271214393,26745019396096,
%U A161010 25239592216021,132269647372591,243274230757741,562881233944576,619036127056621
%N A161010 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 14.
%C A161010 a(n) is the number of lattices L in Z^13 such that the quotient group Z^13 / L is C_n. - _Álvar Ibeas_, Nov 26 2015
%H A161010 Álvar Ibeas, <a href="/A161010/b161010.txt">Table of n, a(n) for n = 1..10000</a>
%H A161010 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 A161010 <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F A161010 a(n) = J_13(n)/J_1(n) where J_13 and J_1(n) = A000010(n) are Jordan functions. - _R. J. Mathar_, Jul 12 2011
%F A161010 From _Álvar Ibeas_, Nov 26 2015: (Start)
%F A161010 Multiplicative with a(p^e) = p^(12e-12) * (p^13-1) / (p-1).
%F A161010 For squarefree n, a(n) = A000203(n^12). (End)
%F A161010 From _Amiram Eldar_, Nov 08 2022: (Start)
%F A161010 Sum_{k=1..n} a(k) ~ c * n^13, where c = (1/13) * Product_{p prime} (1 + (p^12-1)/((p-1)*p^13)) = 0.14949521105... .
%F A161010 Sum_{k>=1} 1/a(k) = zeta(12)*zeta(13) * Product_{p prime} (1 - 2/p^13 + 1/p^25) = 1.0001233729754... . (End)
%F A161010 a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^13). - _Ridouane Oudra_, Apr 02 2025
%p A161010 f:= proc(n) local t; mul(t[1]^(12*t[2]-12)*(t[1]^13-1)/(t[1]-1), t = ifactors(n)[2]) end proc:
%p A161010 seq(f(n),n=1..100); # _Robert Israel_, Dec 08 2015
%t A161010 b = 14; Table[Sum[MoebiusMu[n/d] d^(b - 1), {d, Divisors@ n}]/EulerPhi@ n, {n, 17}] (* _Michael De Vlieger_, Nov 27 2015 *)
%t A161010 f[p_, e_] := p^(12*e - 12) * (p^13-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Nov 08 2022 *)
%o A161010 (PARI) vector(100, n, sumdiv(n^12, d, if(ispower(d, 13), moebius(sqrtnint(d, 13))*sigma(n^12/d), 0))) \\ _Altug Alkan_, Nov 26 2015
%o A161010 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^13 - 1)*f[i,1]^(12*f[i,2] - 12)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022
%Y A161010 Column 13 of A263950.
%Y A161010 Cf. A000010, A013670, A000203, A013671, A160897.
%K A161010 nonn,mult
%O A161010 1,2
%A A161010 _N. J. A. Sloane_, Nov 19 2009
%E A161010 Definition corrected by _Enrique Pérez Herrero_, Oct 30 2010