A344219 Number of cyclic subgroups of the group (C_n)^5, where C_n is the cyclic group of order n.

1, 32, 122, 528, 782, 3904, 2802, 8464, 9923, 25024, 16106, 64416, 30942, 89664, 95404, 135440, 88742, 317536, 137562, 412896, 341844, 515392, 292562, 1032608, 488907, 990144, 803804, 1479456, 732542, 3052928, 954306, 2167056, 1964932, 2839744, 2191164, 5239344, 1926222

Offset

1,2

Comments

Inverse Moebius transform of A160893.

Links

Seiichi Manyama, 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_3|n, x_4|n, x_5|n} phi(x_1)*phi(x_2)*phi(x_3)*phi(x_4)*phi(x_5)/phi(lcm(x_1, x_2, x_3, x_4, x_5)).

If p is prime, a(p) = 1 + (p^5 - 1)/(p - 1).

From Amiram Eldar, Nov 15 2022: (Start)

Multiplicative with a(p^e) = 1 + ((p^5 - 1)/(p - 1))*((p^(4*e) - 1)/(p^4 - 1)).

Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)/5) * Product_{p prime} (1 + 1/p^2 + 1/p^3 + 1/p^4 + 1/p^5) = 0.3939461744... . (End)

Mathematica

f[p_, e_] := 1 + ((p^5 - 1)/(p - 1))*((p^(4*e) - 1)/(p^4 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 15 2022 *)

Prog

(PARI) a(n) = sumdiv(n, i, sumdiv(n, j, sumdiv(n, k, sumdiv(n, l, sumdiv(n, m, eulerphi(i)*eulerphi(j)*eulerphi(k)*eulerphi(l)*eulerphi(m)/eulerphi(lcm([i, j, k, l, m])))))));
(PARI) a160893(n) = sumdiv(n, d, moebius(n/d)*d^5)/eulerphi(n);
a(n) = sumdiv(n, d, a160893(d));

Crossrefs

Cf. A000010, A013663, A060648, A064969, A160893, A280184.

Sequence in context: A203965 A203958 A005903 * A271532 A264480 A247155

Adjacent sequences: A344216 A344217 A344218 * A344220 A344221 A344222

Keyword

nonn,mult

Author

Seiichi Manyama, May 12 2021

Status

approved