cp's OEIS Frontend

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

%I A344304 #22 Nov 15 2022 09:17:15
%S A344304 1,256,3281,32896,97657,839936,960801,4210816,7176641,25000192,
%T A344304 21435889,107931776,67977561,245965056,320412617,538984576,435984841,
%U A344304 1837220096,943531281,3212524672,3152388081,5487587584,3559590241,13815687296,7629472657,17402255616
%N A344304 Number of cyclic subgroups of the group (C_n)^8, where C_n is the cyclic group of order n.
%C A344304 Inverse Moebius transform of A160908.
%H A344304 Amiram Eldar, <a href="/A344304/b344304.txt">Table of n, a(n) for n = 1..10000</a>
%H A344304 László Tóth, <a href="http://arxiv.org/abs/1203.6201">On the number of cyclic subgroups of a finite abelian group</a>, arXiv: 1203.6201 [math.GR], 2012.
%F A344304 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)).
%F A344304 If p is prime, a(p) = 1 + (p^8 - 1)/(p - 1).
%F A344304 From _Amiram Eldar_, Nov 15 2022: (Start)
%F A344304 Multiplicative with a(p^e) = 1 + ((p^8 - 1)/(p - 1))*((p^(7*e) - 1)/(p^7 - 1)).
%F A344304 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)
%t A344304 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 *)
%o A344304 (PARI) a160908(n) = sumdiv(n, d, moebius(n/d)*d^8)/eulerphi(n);
%o A344304 a(n) = sumdiv(n, d, a160908(d));
%Y A344304 Cf. A000010, A013666, A060648, A064969, A160908, A280184, A344219, A344302, A344303, A344305, A344306.
%K A344304 nonn,mult
%O A344304 1,2
%A A344304 _Seiichi Manyama_, May 14 2021