A344201 Number of cyclic subgroups of the group (C_n)^n, where C_n is the cyclic group of order n.

1, 4, 14, 136, 782, 23360, 137258, 4210816, 64576643, 2500000768, 28531167062, 2229573502976, 25239592216022, 1852001137606656, 54736740117685528, 2305878194659557376, 51702516367896047762, 6557734713069408616448, 109912203092239643840222

Offset: 1

Author

Seiichi Manyama, May 12 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..387
• László Tóth, On the number of cyclic subgroups of a finite abelian group, arXiv: 1203.6201 [math.GR], 2012.

Crossrefs

Cf. A060648, A064969, A280184, A344219.

Programs

Mathematica

b[n_, k_] := DivisorSum[n, MoebiusMu[n/#] * #^k &] / EulerPhi[n]; a[n_] := DivisorSum[n, b[#, n] &]; Array[a, 20] (* Amiram Eldar, Oct 04 2023 *)

PARI

b(n, k) = sumdiv(n, d, moebius(n/d)*d^k)/eulerphi(n);
a(n) = sumdiv(n, d, b(d, n));

Formula

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

a(n) = Sum_{d|n} b(d, n), where b(n, k) = ( Sum_{d|n} mu(n/d) * d^k )/phi(n).

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