A344306 Number of cyclic subgroups of the group (C_n)^10, where C_n is the cyclic group of order n.

1, 1024, 29525, 524800, 2441407, 30233600, 47079209, 268698112, 581150417, 2500000768, 2593742461, 15494720000, 11488207655, 48209110016, 72082541675, 137573433856, 125999618779, 595098027008, 340614792101, 1281250393600

Offset

1,2

Comments

Inverse Moebius transform of A160957.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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_10|n} phi(x_1)*phi(x_2)* ... *phi(x_10)/phi(lcm(x_1, x_2, ..., x_10)).

If p is prime, a(p) = 1 + (p^10 - 1)/(p - 1).

From Amiram Eldar, Nov 15 2022: (Start)

Multiplicative with a(p^e) = 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)).

Sum_{k=1..n} a(k) ~ c * n^10, where c = (zeta(10)/10) * Product_{p prime} ((1-1/p^9)/(p^2*(1-1/p))) = 0.1944248708... . (End)

Mathematica

f[p_, e_] := 1 + ((p^10 - 1)/(p - 1))*((p^(9*e) - 1)/(p^9 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)

Prog

(PARI) a160957(n) = sumdiv(n, d, moebius(n/d)*d^10)/eulerphi(n);
a(n) = sumdiv(n, d, a160957(d));

Crossrefs

Cf. A000010, A013668, A160957, A060648, A064969, A280184, A344219, A344302, A344303, A344304, A344305.

Sequence in context: A205611 A016781 A247933 * A205349 A016805 A230790

Adjacent sequences: A344303 A344304 A344305 * A344307 A344308 A344309

Keyword

nonn,mult

Author

Seiichi Manyama, May 14 2021

Status

approved