cp's OEIS Frontend

A280184 Number of cyclic subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.

%I A280184 #47 Jan 24 2024 08:02:43
%S A280184 1,16,41,136,157,656,401,1096,1121,2512,1465,5576,2381,6416,6437,8776,
%T A280184 5221,17936,7241,21352,16441,23440,12721,44936,19657,38096,30281,
%U A280184 54536,25261,102992,30785,70216,60065,83536,62957,152456,52061,115856,97621,172072,70645,263056,81401,199240,175997,203536,106081,359816,137601,314512
%N A280184 Number of cyclic subgroups of the group C_n x C_n x C_n x C_n, where C_n is the cyclic group of order n.
%C A280184 Inverse Moebius transform of A160891. - _Seiichi Manyama_, May 12 2021
%H A280184 Amiram Eldar, <a href="/A280184/b280184.txt">Table of n, a(n) for n = 1..10000</a>
%H A280184 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.
%H A280184 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 A280184 a(n) = Sum_{a|n, b|n, c|n, d|n} phi(a)*phi(b)*phi(c)*phi(d)/phi(lcm(a, b, c, d)), where phi is Euler totient function (cf. A000010).
%F A280184 From _Amiram Eldar_, Nov 15 2022: (Start)
%F A280184 Multiplicative with a(p^e) = 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)).
%F A280184 Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)/4) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4) = 0.5010902655... . (End)
%F A280184 a(n) = Sum_{d divides n} J_4(d)/phi(d) = Sum_{1 <= i, j, k, l <= n} 1/phi(n/gcd(i,j,k,l,n)), where the Jordan totient function J_4(n) = A059377(n). - _Peter Bala_, Jan 23 2024
%p A280184 with(numtheory):
%p A280184 # define Jordan totient function J(r,n)
%p A280184 J(r,n) := add(d^r*mobius(n/d), d in divisors(n)):
%p A280184 seq(add(J(4,d)/phi(d), d in divisors(n)), n = 1..50); # _Peter Bala_, Jan 23 2024
%t A280184 a[n_] := With[{dd = Divisors[n]}, Sum[Times @@ EulerPhi @ {x, y, z, t} / EulerPhi[LCM[x, y, z, t]], {x, dd}, {y, dd}, {z, dd}, {t, dd}]];
%t A280184 Array[a, 50] (* _Jean-François Alcover_, Sep 28 2018 *)
%t A280184 f[p_, e_] := 1 + (p^3 + p^2 + p + 1)*((p^(3*e) - 1)/(p^3 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 15 2022 *)
%o A280184 (PARI) a(n) = sumdiv(n, x, sumdiv(n, y, sumdiv(n, z, sumdiv(n, t, eulerphi(x)*eulerphi(y)*eulerphi(z)*eulerphi(t)/eulerphi(lcm([x, y, z, t])))))); \\ _Michel Marcus_, Feb 26 2018
%o A280184 (PARI) a160891(n) = sumdiv(n, d, moebius(n/d)*d^4)/eulerphi(n);
%o A280184 a(n) = sumdiv(n, d, a160891(d)); \\ _Seiichi Manyama_, May 12 2021
%Y A280184 Cf. A000010, A013662, A060648, A064969, A160891, A280162, A344219, A344302, A344303.
%K A280184 nonn,mult,easy
%O A280184 1,2
%A A280184 _Laszlo Toth_, Dec 28 2016