A160908 a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 9.
1, 255, 3280, 32640, 97656, 836400, 960800, 4177920, 7173360, 24902280, 21435888, 107059200, 67977560, 245004000, 320311680, 534773760, 435984840, 1829206800, 943531280, 3187491840, 3151424000, 5466151440, 3559590240, 13703577600, 7629375000, 17334277800
Offset: 1
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.
Programs
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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 *)
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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
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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
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)
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^8). - Ridouane Oudra, Apr 01 2025
Extensions
Definition corrected by Enrique Pérez Herrero, Oct 28 2010
Comments