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

1,2

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.

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).

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 *)

Prog

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

Crossrefs

Cf. A060648, A064969, A280184, A344219.

Sequence in context: A302021 A006824 A333730 * A254718 A162077 A247526

Adjacent sequences: A344198 A344199 A344200 * A344202 A344203 A344204

Keyword

nonn

Author

Seiichi Manyama, May 12 2021

Status

approved