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URL: https://oeis.org/A344306

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A344306
Number of cyclic subgroups of the group (C_n)^10, where C_n is the cyclic group of order n.
5
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
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));
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 14 2021
STATUS
approved