reviewed

approved

proposed

reviewed

editing

proposed

Cf. A000010, A013664, A013665, A160897.

Amiram Eldar, <a href="/A160913/b160913.txt">Table of n, a(n) for n = 1..10000</a>

255, 32385, 278715, 2072640, 4980405, 35396805, 35000535, 132648960, 203183235, 632511435, 496922835, 2265395520, 1333405965, 4445067945, 5443582665, 8489533440, 6539772585, 25804270845, 12663182955, 40480731840, 38255584755, 63109200045, 39465022215, 144985313280

a(n) = ((2^b-1)/phi(n))*Sum_{d|n} Moebius(n/d)*d^(b-1) for b = 8.

J. H. Kwak and J. 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.

Jin Ho Kwak and Jaeun Lee, <a href="https://doi.org/10.1142/9789812799890_0005">Enumeration of graph coverings, surface branched coverings and related group theory</a>, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.

From Amiram Eldar, Nov 08 2022: (Start)

Sum_{k=1..n} a(k) ~ c * n^7, where c = (255/7) * Product_{p prime} (1 + (p^6-1)/((p-1)*p^7)) = 70.419647503... .

Sum_{k>=1} 1/a(k) = (zeta(6)*zeta(7)/255) * Product_{p prime} (1 - 2/p^7 + 1/p^13) = 0.003956793297... . (End)

f[p_, e_] := p^(6*e - 6) * (p^7-1) / (p-1); a[1] = 255; a[n_] := 255 * Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Nov 08 2022 *)

(PARI) a(n) = {my(f = factor(n)); 255 * prod(i = 1, #f~, (f[i, 1]^7 - 1)*f[i, 1]^(6*f[i, 2] - 6)/(f[i, 1] - 1)); } \\ Amiram Eldar, Nov 08 2022

Cf. A013664, A013665, A160897.

approved

editing

editing

approved

a(n) = 255*A160897(n). - R. J. Mathar, Mar 15 2016

approved

editing

_N. J. A. Sloane (njas(AT)research.att.com), _, Nov 19 2009