OFFSET
1,2
COMMENTS
Sum of the 10th 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^10 * mu(n/d)^2.
a(n) = n^10 * Sum_{d|n} mu(d)^2 / d^10.
Multiplicative with a(p^e) = p^(10*e) + p^(10*e-10). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-10)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^11 * zeta(11) / (11 * zeta(22)) = 1222532449149375 * n^11 * zeta(11) / (155366 * Pi^22)}.
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^10/(p^20-1)) = 1.000993621149252443797467720671490169127513829380371486971107300011796... (End)
MATHEMATICA
f[p_, e_] := p^(10*e) + p^(10*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* Amiram Eldar, Feb 08 2022 *)
PROG
(PARI) a(n)=sumdiv(n, d, moebius(n/d)^2*d^10);
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^10*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Wesley Ivan Hurt, Feb 06 2022
STATUS
approved