A344303 - OEIS

%I #22 Nov 15 2022 09:17:41

%S 1,128,1094,8256,19532,140032,137258,528448,797891,2500096,1948718,

%T 9032064,5229044,17569024,21368008,33820736,25646168,102130048,

%U 49659542,161256192,150160252,249435904,154764794,578122112,305191407,669317632,581662904,1133202048

%N Number of cyclic subgroups of the group (C_n)^7, where C_n is the cyclic group of order n.

%C Inverse Moebius transform of A160897.

%H Amiram Eldar, <a href="/A344303/b344303.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 a(n) = Sum_{x_1|n, x_2|n, ..., x_7|n} phi(x_1)*phi(x_2)* ... *phi(x_7)/phi(lcm(x_1, x_2, ..., x_7)).

%F If p is prime, a(p) = 1 + (p^7 - 1)/(p - 1).

%F From _Amiram Eldar_, Nov 15 2022: (Start)

%F Multiplicative with a(p^e) = 1 + ((p^7 - 1)/(p - 1))*((p^(6*e) - 1)/(p^6 - 1)).

%F Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)/7) * Product_{p prime} ((1-1/p^6)/(p^2*(1-1/p))) = 0.2784611791... . (End)

%t f[p_, e_] := 1 + ((p^7 - 1)/(p - 1))*((p^(6*e) - 1)/(p^6 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* _Amiram Eldar_, Nov 15 2022 *)

%o (PARI) a160897(n) = sumdiv(n, d, moebius(n/d)*d^7)/eulerphi(n);

%o a(n) = sumdiv(n, d, a160897(d));

%Y Cf. A000010, A013665, A060648, A064969, A280184, A344219, A344302, A344304, A344305, A344306.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 14 2021