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%I #35 Nov 08 2022 08:07:19
%S 1,31,121,496,781,3751,2801,7936,9801,24211,16105,60016,30941,86831,
%T 94501,126976,88741,303831,137561,387376,338921,499255,292561,960256,
%U 488125,959171,793881,1389296,732541,2929531,954305,2031616,1948705
%N a(n) = Sum_{d|n} Möbius(n/d)*d^5/phi(n).
%C a(n) is the number of lattices L in Z^5 such that the quotient group Z^5 / L is C_nm x (C_m)^4 (and also (C_nm)^4 x C_m), for every m>=1. - _Álvar Ibeas_, Oct 30 2015
%H Enrique Pérez Herrero, <a href="/A160893/b160893.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_5(n)/J_1(n) = J_5(n)/phi(n) = A059378(n)/A000010(n), where J_k is the k-th Jordan totient function. - _Enrique Pérez Herrero_, Oct 19 2010
%F Multiplicative with a(p^e) = p^(4e-4)*(1 + p+ p^2 + p^3 + p^4). - _R. J. Mathar_, Jul 10 2011
%F For squarefree n, a(n) = A000203(n^4). - _Álvar Ibeas_, Oct 30 2015
%F From _Amiram Eldar_, Nov 08 2022: (Start)
%F Sum_{k=1..n} a(k) ~ c * n^5, where c = (1/5) * Product_{p prime} (1 + (p^4-1)/((p-1)*p^5)) = 0.3799167034... .
%F Sum_{k>=1} 1/a(k) = zeta(4)*zeta(5) * Product_{p prime} (1 - 2/p^5 + 1/p^9) = 1.0449010968... . (End)
%p A160893 := proc(n) a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; a := a*p^(4*e-4)*(1+p+p^2+p^3+p^4) ; end do; a; end proc: # _R. J. Mathar_, Jul 10 2011
%t A160893[n_]:=DivisorSum[n,MoebiusMu[n/# ]*#^(6-1)/EulerPhi[n]&] (* _Enrique Pérez Herrero_, Oct 19 2010 *)
%t f[p_, e_] := p^(4*e - 4)*(1 + p + p^2 + p^3 + p^4); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* _Amiram Eldar_, Nov 08 2022 *)
%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d^5/eulerphi(n)); \\ _Michel Marcus_, Feb 15 2015
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^5 - 1)*f[i, 1]^(4*f[i, 2] - 4)/(f[i, 1] - 1)); } \\ _Amiram Eldar_, Nov 08 2022
%Y Column 5 of A263950.
%Y Cf. A000010, A000203, A013662, A013663, A059378.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_, Nov 19 2009
%E Definition corrected by _Enrique Pérez Herrero_, Oct 19 2010
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