OFFSET
1,2
COMMENTS
a(n) is the number of lattices L in Z^14 such that the quotient group Z^14 / L is C_n. - Álvar Ibeas, Nov 26 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_14(n)/J_1(n) where J_14 and J_1(n) = A000010(n) are Jordan functions. - R. J. Mathar, Jul 12 2011
From Álvar Ibeas, Nov 26 2015: (Start)
Multiplicative with a(p^e) = p^(13e-13) * (p^14-1) / (p-1).
For squarefree n, a(n) = A000203(n^13). (End)
From Amiram Eldar, Nov 08 2022: (Start)
Sum_{k=1..n} a(k) ~ c * n^14, where c = (1/14) * Product_{p prime} (1 + (p^13-1)/((p-1)*p^14)) = 0.1388226555... .
Sum_{k>=1} 1/a(k) = zeta(13)*zeta(14) * Product_{p prime} (1 - 2/p^14 + 1/p^27) = 1.00006146517418... . (End)
MAPLE
A161025 := proc(n)
add(numtheory[mobius](n/d)*d^14, d=numtheory[divisors](n)) ;
%/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
printf("%d %d\n", n, A161025(n)) ;
end do: # R. J. Mathar, Mar 15 2016
MATHEMATICA
A161025[n_]:=DivisorSum[n, MoebiusMu[n/#]*#^(15-1)/EulerPhi[n]&]
f[p_, e_] := p^(13*e - 13) * (p^14-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)
PROG
(PARI) vector(100, n, sumdiv(n^13, d, if(ispower(d, 14), moebius(sqrtnint(d, 14))*sigma(n^13/d), 0))) \\ Altug Alkan, Nov 26 2015
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^14 - 1)*f[i, 1]^(13*f[i, 2] - 13)/(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 30 2010
STATUS
approved