A344305 Number of cyclic subgroups of the group (C_n)^9, where C_n is the cyclic group of order n.

1, 512, 9842, 131328, 488282, 5039104, 6725602, 33620224, 64576643, 250000384, 235794770, 1292530176, 883708282, 3443508224, 4805671444, 8606777600, 7411742282, 33063241216, 17927094322, 64125098496, 66193374884, 120726922240, 81870575522, 330890244608

Offset

1,2

Comments

Inverse Moebius transform of A160953.

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_9|n} phi(x_1)*phi(x_2)* ... *phi(x_9)/phi(lcm(x_1, x_2, ..., x_9)).

If p is prime, a(p) = 1 + (p^9 - 1)/(p - 1).

From Amiram Eldar, Nov 15 2022: (Start)

Multiplicative with a(p^e) = 1 + ((p^9 - 1)/(p - 1))*((p^(8*e) - 1)/(p^8 - 1)).

Sum_{k=1..n} a(k) ~ c * n^9, where c = (zeta(9)/9) * Product_{p prime} ((1-1/p^8)/(p^2*(1-1/p))) = 0.2161023934... . (End)

Mathematica

f[p_, e_] := 1 + ((p^9 - 1)/(p - 1))*((p^(8*e) - 1)/(p^8 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)

Prog

(PARI) a160953(n) = sumdiv(n, d, moebius(n/d)*d^9)/eulerphi(n);
a(n) = sumdiv(n, d, a160953(d));

Crossrefs

Cf. A000010, A013667, A160953, A060648, A064969, A280184, A344219, A344302, A344303, A344304, A344306.

Sequence in context: A255028 A247932 A090007 * A254797 A254837 A254560

Adjacent sequences: A344302 A344303 A344304 * A344306 A344307 A344308

Keyword

nonn,mult

Author

Seiichi Manyama, May 14 2021

Status

approved