A160953 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 10.

1, 511, 9841, 130816, 488281, 5028751, 6725601, 33488896, 64566801, 249511591, 235794769, 1287360256, 883708281, 3436782111, 4805173321, 8573157376, 7411742281, 32993635311, 17927094321, 63874967296, 66186639441, 120491126959, 81870575521, 329564225536

Offset: 1

Author

N. J. A. Sloane, Nov 19 2009

Keywords

• nonn
• mult

Comments

a(n) is the number of lattices L in Z^9 such that the quotient group Z^9 / L is C_n. - Álvar Ibeas, Nov 03 2015

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
• Enrique Pérez Herrero, Mathematica Package: Jordan Totient Function.
• 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 9 of A263950.

Cf. A000010, A000203, A008683, A013666, A013667, A069094.

Programs

Maple

A160953 := proc(n)
    add(numtheory[mobius](n/d)*d^9,d=numtheory[divisors](n)) ;
    %/numtheory[phi](n) ;
end proc:
for n from 1 to 5000 do
    printf("%d %d\n",n,A160953(n)) ;
end do: # R. J. Mathar, Mar 14 2016

Mathematica

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A160953[n_]:=JordanTotient[n,9]/JordanTotient[n];
f[p_, e_] := p^(8*e - 8) * (p^9-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

PARI

vector(100, n, sumdiv(n^8, d, if(ispower(d, 9), moebius(sqrtnint(d, 9))*sigma(n^8/d), 0))) \\ Altug Alkan, Nov 05 2015

PARI

a(n) = {f = factor(n); for (i=1, #f~, p = f[i,1]; f[i,1] = p^(8*f[i,2]-8)*(p^9-1)/(p-1); f[i,2] = 1;); factorback(f);} \\ Michel Marcus, Nov 12 2015

Formula

a(n) = J_9(n)/phi(n) = A069094(n)/A000010(n).

From Álvar Ibeas, Nov 03 2015: (Start)

Multiplicative with a(p^e) = p^(8e-8) * (p^9-1) / (p-1).

For squarefree n, a(n) = A000203(n^8). (End)

From Amiram Eldar, Nov 08 2022: (Start)

Sum_{k=1..n} a(k) ~ c * n^9, where c = (1/9) * Product_{p prime} (1 + (p^8-1)/((p-1)*p^9)) = 0.2156692448... .

Sum_{k>=1} 1/a(k) = zeta(8)*zeta(9) * Product_{p prime} (1 - 2/p^9 + 1/p^17) = 1.002068659133... . (End)

a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^9). - Ridouane Oudra, Apr 01 2025

Extensions

Definition corrected by Enrique Pérez Herrero, Oct 30 2010