OFFSET
1,2
COMMENTS
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
LINKS
Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
Jin Ho Kwak and Jaeun Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
FORMULA
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
Multiplicative with a(p^e) = p^(4e-4)*(1 + p+ p^2 + p^3 + p^4). - R. J. Mathar, Jul 10 2011
For squarefree n, a(n) = A000203(n^4). - Álvar Ibeas, Oct 30 2015
From Amiram Eldar, Nov 08 2022: (Start)
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... .
Sum_{k>=1} 1/a(k) = zeta(4)*zeta(5) * Product_{p prime} (1 - 2/p^5 + 1/p^9) = 1.0449010968... . (End)
MAPLE
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
MATHEMATICA
A160893[n_]:=DivisorSum[n, MoebiusMu[n/# ]*#^(6-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 19 2010 *)
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 *)
PROG
(PARI) a(n) = sumdiv(n, d, moebius(n/d)*d^5/eulerphi(n)); \\ Michel Marcus, Feb 15 2015
(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
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
N. J. A. Sloane, Nov 19 2009
EXTENSIONS
Definition corrected by Enrique Pérez Herrero, Oct 19 2010
STATUS
approved