OFFSET
1,2
COMMENTS
Sum of the 9th powers of the divisor complements of the squarefree divisors of n.
LINKS
Sebastian Karlsson, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{d|n} d^9 * mu(n/d)^2.
a(n) = n^9 * Sum_{d|n} mu(d)^2 / d^9.
Multiplicative with a(p^e) = p^(9*e) + p^(9*e-9). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-9)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^10 * zeta(10) / (10 * zeta(20)) = 3273645375 * n^10 / (349222 * Pi^10).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^9/(p^18-1)) = 1.002004575331916689985388864168116922608947780516939765639888137700557... (End)
MATHEMATICA
f[p_, e_] := p^(9*e) + p^(9*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Feb 08 2022 *)
PROG
(PARI) a(n)=sumdiv(n, d, moebius(n/d)^2*d^9);
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^9*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022
(Python)
from math import prod
from sympy import factorint
def A351304(n): return prod(p**(9*e)+p**(9*(e-1)) for p, e in factorint(n).items()) # Chai Wah Wu, Sep 28 2024
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Wesley Ivan Hurt, Feb 06 2022
STATUS
approved