A161167 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 17.

1, 65535, 21523360, 2147450880, 38146972656, 1410533397600, 5538821761600, 70367670435840, 308836690967520, 2499961853010960, 4594972986357216, 46220358372556800, 55451384098598320, 362986684146456000, 821051025385244160, 2305807824841605120, 3041324492229179280

Offset: 1

Author

N. J. A. Sloane, Nov 19 2009

Keywords

• nonn
• mult

Comments

a(n) is the number of lattices L in Z^16 such that the quotient group Z^16 / 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.
• Index to Jordan function ratios J_k/J_1.

Crossrefs

Column 16 of A263950.

Cf. A000010, A000203, A008683, A013673, A013674.

Programs

Maple

A161167 := proc(n)
    add(numtheory[mobius](n/d)*d^16,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A161167(n)) ;
end do: # R. J. Mathar, Mar 15 2016

Mathematica

A161167[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(17-1)/EulerPhi[n]&]; Array[A161167,20]
f[p_, e_] := p^(15*e - 15) * (p^16-1) / (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^15, d, if(ispower(d, 16), moebius(sqrtnint(d, 16))*sigma(n^15/d), 0))) \\ Altug Alkan, Nov 26 2015

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^16 - 1)*f[i,1]^(15*f[i,2] - 15)/(f[i,1] - 1));} \\ Amiram Eldar, Nov 08 2022

Formula

a(n) = J_16(n)/J_1(n) = J_16(n)/A000010(n), where J_k is the k-th Jordan totient function.

From Álvar Ibeas, Nov 26 2015: (Start)

Multiplicative with a(p^e) = p^(15e-15) * (p^16-1) / (p-1).

For squarefree n, a(n) = A000203(n^15). (End)

From Amiram Eldar, Nov 08 2022: (Start)

Sum_{k=1..n} a(k) ~ c * n^16, where c = (1/16) * Product_{p prime} (1 + (p^15-1)/((p-1)*p^16)) = 0.1214735403... .

Sum_{k>=1} 1/a(k) = zeta(15)*zeta(16) * Product_{p prime} (1 - 2/p^16 + 1/p^31) = 1.00001530597583... . (End)

a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^16). - Ridouane Oudra, Apr 02 2025

Extensions

Definition corrected by Enrique Pérez Herrero, Oct 30 2010