A161213 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 18.

1, 131071, 64570081, 8589869056, 190734863281, 8463265086751, 38771752331201, 562945658454016, 2779530261754401, 24999809265103951, 50544702849929377, 554648540725313536, 720867993281778161, 5081852349802846271

Offset

1,2

Comments

a(n) is the number of lattices L in Z^17 such that the quotient group Z^17 / 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.

Formula

a(n) = J_17(n)/A000010(n), where J_17 is the 17th Jordan totient function.

From Álvar Ibeas, Nov 26 2015: (Start)

Multiplicative with a(p^e) = p^(16e-16) * (p^17-1) / (p-1).

For squarefree n, a(n) = A000203(n^16). (End)

From Amiram Eldar, Nov 08 2022: (Start)

Sum_{k=1..n} a(k) ~ c * n^17, where c = (1/17) * Product_{p prime} (1 + (p^16-1)/((p-1)*p^17)) = 0.1143286202... .

Sum_{k>=1} 1/a(k) = zeta(16)*zeta(17) * Product_{p prime} (1 - 2/p^17 + 1/p^33) = 1.000007645061593... . (End)

Maple

A161213 := proc(n)
    add(numtheory[mobius](n/d)*d^17, d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n", n, A161213(n)) ;
end do: # R. J. Mathar, Mar 15 2016

Mathematica

A161213[n_]:=DivisorSum[n, MoebiusMu[n/#]*#^(18-1)/EulerPhi[n]&]; Array[A161213, 20]
f[p_, e_] := p^(16*e - 16) * (p^17-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

Prog

(PARI) A161213(n)=sumdiv(n, d, moebius(n/d)*d^17)/eulerphi(n);
(PARI) vector(100, n, sumdiv(n^16, d, if(ispower(d, 17), moebius(sqrtnint(d, 16))*sigma(n^16/d), 0))) \\ Altug Alkan, Nov 26 2015
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^17 - 1)*f[i, 1]^(16*f[i, 2] - 16)/(f[i, 1] - 1)); } \\ Amiram Eldar, Nov 08 2022

Crossrefs

Column 17 of A263950.

Cf. A000010, A000203, A013674, A013675.

Sequence in context: A051331 A075972 A011567 * A075967 A075968 A022533

Adjacent sequences: A161210 A161211 A161212 * A161214 A161215 A161216

Keyword

nonn,mult

Author

N. J. A. Sloane, Nov 19 2009

Extensions

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

Status

approved