%I #40 Nov 08 2022 08:07:23
%S 1,63,364,2016,3906,22932,19608,64512,88452,246078,177156,733824,
%T 402234,1235304,1421784,2064384,1508598,5572476,2613660,7874496,
%U 7137312,11160828,6728904,23482368,12206250,25340742,21493836,39529728,21243690,89572392
%N a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 7.
%C a(n) is the number of lattices L in Z^6 such that the quotient group Z^6 / L is C_nm x (C_m)^5 (and also (C_nm)^5 x C_m), for every m>=1. - _Álvar Ibeas_, Oct 30 2015
%H Enrique Pérez Herrero, <a href="/A160895/b160895.txt">Table of n, a(n) for n = 1..5000</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_6(n)/J_1(n)=J_6(n)/phi(n)=A069091(n)/A000010(n), where J_k is the k-th Jordan totient function. - _Enrique Pérez Herrero_, Oct 20 2010
%F Multiplicative with a(p^e) = p^(5e-5)*(1+p+p^2+p^3+p^4+p^5). - _R. J. Mathar_, Jul 10 2011
%F For squarefree n, a(n) = A000203(n^5). - _Álvar Ibeas_, Oct 30 2015
%F From _Amiram Eldar_, Nov 08 2022: (Start)
%F Sum_{k=1..n} a(k) ~ c * n^6, where c = (1/6) * Product_{p prime} (1 + (p^5-1)/((p-1)*p^6)) = 0.3203646372... .
%F Sum_{k>=1} 1/a(k) = zeta(5)*zeta(6) * Product_{p prime} (1 - 2/p^6 + 1/p^11) = 1.0195114923... . (End)
%p A160895 := proc(n) a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; a := a*p^(5*e-5)*(1+p+p^2+p^3+p^4+p^5) ; end do; a; end proc: # _R. J. Mathar_, Jul 10 2011
%t A160895[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(7-1)/EulerPhi[n]&] (* _Enrique Pérez Herrero_, Oct 20 2010 *)
%t f[p_, e_] := p^(5*e - 5) * (p^6-1) / (p-1); ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* _Amiram Eldar_, Nov 08 2022 *)
%o (PARI) vector(50, n, sumdiv(n^5, d, if(ispower(d, 6), moebius(sqrtnint(d, 6))*sigma(n^5/d), 0))) \\ _Altug Alkan_, Oct 30 2014
%o (PARI) a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i,1] = p^(5*f[i,2]-5)*(1+p+p^2+p^3+p^4+p^5); f[i,2] = 1;); factorback(f);} \\ _Michel Marcus_, Nov 12 2015
%Y Column 6 of A263950.
%Y Cf. A000010, A000203, A013663, A013664, A069091.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_, Nov 19 2009
%E Definition corrected by _Enrique Pérez Herrero_, Oct 20 2010