cp's OEIS Frontend

A344306 Number of cyclic subgroups of the group (C_n)^10, where C_n is the cyclic group of order n.

%I A344306 #21 Nov 15 2022 09:17:09
%S A344306 1,1024,29525,524800,2441407,30233600,47079209,268698112,581150417,
%T A344306 2500000768,2593742461,15494720000,11488207655,48209110016,
%U A344306 72082541675,137573433856,125999618779,595098027008,340614792101,1281250393600
%N A344306 Number of cyclic subgroups of the group (C_n)^10, where C_n is the cyclic group of order n.
%C A344306 Inverse Moebius transform of A160957.
%H A344306 Amiram Eldar, <a href="/A344306/b344306.txt">Table of n, a(n) for n = 1..10000</a>
%H A344306 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 A344306 a(n) = Sum_{x_1|n, x_2|n, ..., x_10|n} phi(x_1)*phi(x_2)* ... *phi(x_10)/phi(lcm(x_1, x_2, ..., x_10)).
%F A344306 If p is prime, a(p) = 1 + (p^10 - 1)/(p - 1).
%F A344306 From _Amiram Eldar_, Nov 15 2022: (Start)
%F A344306 Multiplicative with a(p^e) = 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)).
%F A344306 Sum_{k=1..n} a(k) ~ c * n^10, where c = (zeta(10)/10) * Product_{p prime} ((1-1/p^9)/(p^2*(1-1/p))) = 0.1944248708... . (End)
%t A344306 f[p_, e_] := 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* _Amiram Eldar_, Nov 15 2022 *)
%o A344306 (PARI) a160957(n) = sumdiv(n, d, moebius(n/d)*d^10)/eulerphi(n);
%o A344306 a(n) = sumdiv(n, d, a160957(d));
%Y A344306 Cf. A000010, A013668, A160957, A060648, A064969, A280184, A344219, A344302, A344303, A344304, A344305.
%K A344306 nonn,mult
%O A344306 1,2
%A A344306 _Seiichi Manyama_, May 14 2021