A344304 Number of cyclic subgroups of the group (C_n)^8, where C_n is the cyclic group of order n.

1, 256, 3281, 32896, 97657, 839936, 960801, 4210816, 7176641, 25000192, 21435889, 107931776, 67977561, 245965056, 320412617, 538984576, 435984841, 1837220096, 943531281, 3212524672, 3152388081, 5487587584, 3559590241, 13815687296, 7629472657, 17402255616

Offset

1,2

Comments

Inverse Moebius transform of A160908.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Formula

a(n) = Sum_{x_1|n, x_2|n, ..., x_8|n} phi(x_1)*phi(x_2)* ... *phi(x_8)/phi(lcm(x_1, x_2, ..., x_8)).

If p is prime, a(p) = 1 + (p^8 - 1)/(p - 1).

From Amiram Eldar, Nov 15 2022: (Start)

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

Sum_{k=1..n} a(k) ~ c * n^8, where c = (zeta(8)/8) * Product_{p prime} ((1-1/p^7)/(p^2*(1-1/p))) = 0.2432888374... . (End)

Mathematica

f[p_, e_] := 1 + ((p^8 - 1)/(p - 1))*((p^(7*e) - 1)/(p^7 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)

Prog

(PARI) a160908(n) = sumdiv(n, d, moebius(n/d)*d^8)/eulerphi(n);
a(n) = sumdiv(n, d, a160908(d));

Crossrefs

Cf. A000010, A013666, A060648, A064969, A160908, A280184, A344219, A344302, A344303, A344305, A344306.

Sequence in context: A236130 A236254 A236251 * A070056 A185575 A185933

Adjacent sequences: A344301 A344302 A344303 * A344305 A344306 A344307

Keyword

nonn,mult

Author

Seiichi Manyama, May 14 2021

Status

approved