OFFSET
1,2
COMMENTS
Inverse Moebius transform of A160908.
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_8|n} phi(x_1)*phi(x_2)* ... *phi(x_8)/phi(lcm(x_1, x_2, ..., x_8)).
If p is prime, a(p) = 1 + (p^8 - 1)/(p - 1).
From Amiram Eldar, Nov 15 2022: (Start)
Multiplicative with a(p^e) = 1 + ((p^8 - 1)/(p - 1))*((p^(7*e) - 1)/(p^7 - 1)).
Sum_{k=1..n} a(k) ~ c * n^8, where c = (zeta(8)/8) * Product_{p prime} ((1-1/p^7)/(p^2*(1-1/p))) = 0.2432888374... . (End)
MATHEMATICA
f[p_, e_] := 1 + ((p^8 - 1)/(p - 1))*((p^(7*e) - 1)/(p^7 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Nov 15 2022 *)
PROG
(PARI) a160908(n) = sumdiv(n, d, moebius(n/d)*d^8)/eulerphi(n);
a(n) = sumdiv(n, d, a160908(d));
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 14 2021
STATUS
approved