cp's OEIS Frontend

A064969 Number of cyclic subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).

%I A064969 #48 Jan 24 2024 08:02:48
%S A064969 1,8,14,36,32,112,58,148,131,256,134,504,184,464,448,596,308,1048,382,
%T A064969 1152,812,1072,554,2072,807,1472,1184,2088,872,3584,994,2388,1876,
%U A064969 2464,1856,4716,1408,3056,2576,4736,1724,6496,1894,4824,4192,4432,2258,8344
%N A064969 Number of cyclic subgroups of the group C_n X C_n X C_n (where C_n is the cyclic group of order n).
%C A064969 Inverse Moebius transform of A160889. - _Vladeta Jovovic_, Nov 21 2009
%H A064969 Vincenzo Librandi, <a href="/A064969/b064969.txt">Table of n, a(n) for n = 1..1000</a>
%H A064969 László Tóth, <a href="http://www.seminariomatematico.polito.it/rendiconti/69-1/97.pdf">Menon's identity and arithmetical sums representing functions of several variables</a>, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110 and <a href="http://arxiv.org/abs/1103.5861">arXiv</a>, arXiv:1103.5861 [math.NT], 2011.
%H A064969 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 A064969 a(n) = Sum_{i|n, j|n, k|n} phi(i)*phi(j)*phi(k)/phi(lcm(i, j, k)), where phi is Euler totient function (cf. A000010).
%F A064969 From _Amiram Eldar_, Nov 15 2022: (Start)
%F A064969 Multiplicative with a(p^e) = 1 + (p^2 + p + 1)*((p^(2*e) - 1)/(p^2 - 1)).
%F A064969 Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^2 + 1/p^3) = A002117 * A330595 / 3 = 0.700772... . (End)
%F A064969 a(n) = Sum_{d divides n} J_3(d)/phi(d) = Sum_{1 <= i, j, k <= n} 1/phi(n/gcd(i,j,k,n)), where the Jordan totient function J_3(n) = A059376(n). - _Peter Bala_, Jan 23 2024
%p A064969 with(numtheory):
%p A064969 # define Jordan totient function J(r,n)
%p A064969 J(r,n) := add(d^r*mobius(n/d), d in divisors(n)):
%p A064969 seq(add(J(3,d)/phi(d), d in divisors(n)), n = 1..50); # _Peter Bala_, Jan 23 2024
%t A064969 a[n_] := Sum[EulerPhi[i] EulerPhi[j] (EulerPhi[k] / EulerPhi[LCM[i, j, k]]), {i, Divisors[n]}, {j, Divisors[n]}, {k, Divisors[n]}];
%t A064969 Array[a, 48] (* _Jean-François Alcover_, Dec 13 2018, after _Vladeta Jovovic_ *)
%t A064969 f[p_, e_] := 1 + (p^2 + p + 1)*((p^(2*e) - 1)/(p^2 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 15 2022 *)
%o A064969 (PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, eulerphi(i)*eulerphi(j)*eulerphi(k)/eulerphi(lcm(lcm(i, j), k))))); \\ _Michel Marcus_, Dec 14 2018
%o A064969 (PARI) a160889(n) = sumdiv(n, d, moebius(n/d)*d^3)/eulerphi(n);
%o A064969 a(n) = sumdiv(n, d, a160889(d)); \\ _Seiichi Manyama_, May 12 2021
%Y A064969 Cf. A000010, A002117, A064803, A060648, A060724, A064767, A160889, A280184, A330595, A344219, A344302, A344303.
%K A064969 nonn,mult,easy
%O A064969 1,2
%A A064969 Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 30 2001
%E A064969 Formula and more terms from _Vladeta Jovovic_, Oct 30 2001