cp's OEIS Frontend

A344305 Number of cyclic subgroups of the group (C_n)^9, where C_n is the cyclic group of order n.

%I A344305 #23 Nov 15 2022 09:17:45
%S A344305 1,512,9842,131328,488282,5039104,6725602,33620224,64576643,250000384,
%T A344305 235794770,1292530176,883708282,3443508224,4805671444,8606777600,
%U A344305 7411742282,33063241216,17927094322,64125098496,66193374884,120726922240,81870575522,330890244608
%N A344305 Number of cyclic subgroups of the group (C_n)^9, where C_n is the cyclic group of order n.
%C A344305 Inverse Moebius transform of A160953.
%H A344305 Amiram Eldar, <a href="/A344305/b344305.txt">Table of n, a(n) for n = 1..10000</a>
%H A344305 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 A344305 a(n) = Sum_{x_1|n, x_2|n, ..., x_9|n} phi(x_1)*phi(x_2)* ... *phi(x_9)/phi(lcm(x_1, x_2, ..., x_9)).
%F A344305 If p is prime, a(p) = 1 + (p^9 - 1)/(p - 1).
%F A344305 From _Amiram Eldar_, Nov 15 2022: (Start)
%F A344305 Multiplicative with a(p^e) = 1 + ((p^9 - 1)/(p - 1))*((p^(8*e) - 1)/(p^8 - 1)).
%F A344305 Sum_{k=1..n} a(k) ~ c * n^9, where c = (zeta(9)/9) * Product_{p prime} ((1-1/p^8)/(p^2*(1-1/p))) = 0.2161023934... . (End)
%t A344305 f[p_, e_] := 1 + ((p^9 - 1)/(p - 1))*((p^(8*e) - 1)/(p^8 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* _Amiram Eldar_, Nov 15 2022 *)
%o A344305 (PARI) a160953(n) = sumdiv(n, d, moebius(n/d)*d^9)/eulerphi(n);
%o A344305 a(n) = sumdiv(n, d, a160953(d));
%Y A344305 Cf. A000010, A013667, A160953, A060648, A064969, A280184, A344219, A344302, A344303, A344304, A344306.
%K A344305 nonn,mult
%O A344305 1,2
%A A344305 _Seiichi Manyama_, May 14 2021