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

Offset

1,2

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.

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)

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 *)

Prog

(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

Crossrefs

Column 9 of A263950.

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

Sequence in context: A204739 A075948 A011559 * A038996 A068025 A075943

Adjacent sequences: A160950 A160951 A160952 * A160954 A160955 A160956

Keyword

nonn,mult

Author

N. J. A. Sloane, Nov 19 2009

Extensions

Definition corrected by Enrique Pérez Herrero, Oct 30 2010

Status

approved