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A160908 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 9. 4
1, 255, 3280, 32640, 97656, 836400, 960800, 4177920, 7173360, 24902280, 21435888, 107059200, 67977560, 245004000, 320311680, 534773760, 435984840, 1829206800, 943531280, 3187491840, 3151424000, 5466151440, 3559590240, 13703577600 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is the number of lattices L in Z^8 such that the quotient group Z^8 / L is C_n. - Álvar Ibeas, Oct 30 2015

LINKS

G. C. Greubel, 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_8(n)/J_1(n) = J_8(n)/phi(n) = A069093(n)/A000010(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Oct 28 2010

From Álvar Ibeas, Oct 30 2015: (Start)

Multiplicative with a(p^e) = p^(7e-7) * (p^8-1) / (p-1).

For squarefree n, a(n) = A000203(n^7). (End)

From Amiram Eldar, Nov 08 2022: (Start)

Sum_{k=1..n} a(k) ~ c * n^8, where c = (1/8) * Product_{p prime} (1 + (p^7-1)/((p-1)*p^8)) = 0.2423008904... .

Sum_{k>=1} 1/a(k) = zeta(7)*zeta(8) * Product_{p prime} (1 - 2/p^8 + 1/p^15) = 1.004270064601... . (End)

MATHEMATICA

A160908[n_]:=DivisorSum[n, MoebiusMu[n/# ]*#^(9-1)/EulerPhi[n]&] (* Enrique Pérez Herrero, Oct 28 2010 *)

f[p_, e_] := p^(7*e - 7) * (p^8-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Nov 08 2022 *)

PROG

(PARI) vector(30, n, sumdiv(n^7, d, if(ispower(d, 8), moebius(sqrtnint(d, 8))*sigma(n^7/d), 0))) \\ Altug Alkan, Oct 30 2015

(PARI) a(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; f[i, 1] = p^(7*f[i, 2]-7)*(p^8-1)/(p-1); f[i, 2] = 1; ); factorback(f); } \\ Michel Marcus, Nov 12 2015

CROSSREFS

Column 8 of A263950.

Cf. A000010, A000203, A013665, A013666, A069093.

Sequence in context: A259247 A204738 A206048 * A038995 A068024 A028524

Adjacent sequences: A160905 A160906 A160907 * A160909 A160910 A160911

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane, Nov 19 2009

EXTENSIONS

Definition corrected by Enrique Pérez Herrero, Oct 28 2010

STATUS

approved

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Last modified November 30 18:07 EST 2022. Contains 358453 sequences. (Running on oeis4.)