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

1, 8191, 797161, 33550336, 305175781, 6529545751, 16148168401, 137422176256, 423644039001, 2499694822171, 3452271214393, 26745019396096, 25239592216021, 132269647372591, 243274230757741, 562881233944576, 619036127056621

Offset: 1

Author

N. J. A. Sloane, Nov 19 2009

Keywords

• nonn
• mult

Comments

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

Links

• Álvar Ibeas, Table of n, a(n) for n = 1..10000
• 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.
• Index to Jordan function ratios J_k/J_1.

Crossrefs

Column 13 of A263950.

Cf. A000010, A013670, A000203, A013671, A160897.

Programs

Maple

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:
seq(f(n),n=1..100); # Robert Israel, Dec 08 2015

Mathematica

b = 14; Table[Sum[MoebiusMu[n/d] d^(b - 1), {d, Divisors@ n}]/EulerPhi@ n, {n, 17}] (* Michael De Vlieger, Nov 27 2015 *)
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 *)

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

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

Formula

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

From Álvar Ibeas, Nov 26 2015: (Start)

Multiplicative with a(p^e) = p^(12e-12) * (p^13-1) / (p-1).

For squarefree n, a(n) = A000203(n^12). (End)

From Amiram Eldar, Nov 08 2022: (Start)

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... .

Sum_{k>=1} 1/a(k) = zeta(12)*zeta(13) * Product_{p prime} (1 - 2/p^13 + 1/p^25) = 1.0001233729754... . (End)

a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^13). - Ridouane Oudra, Apr 02 2025

Extensions

Definition corrected by Enrique Pérez Herrero, Oct 30 2010