OFFSET
1,2
COMMENTS
Sum of the 6th 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^6 * mu(n/d)^2.
a(n) = n^6 * Sum_{d|n} mu(d)^2 / d^6.
Multiplicative with a(p^e) = p^(6*e) + p^(6*e-6). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^7 * zeta(7) / (7 * zeta(14)) = 2606175 * n^7 * zeta(7) / (2 * Pi^14).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^6/(p^12-1)) = 1.01709659289559607702424749979498914920118274875188346777424441790304... (End)
MATHEMATICA
f[p_, e_] := p^(6*e) + p^(6*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Feb 08 2022 *)
PROG
(PARI) a(n)=sumdiv(n, d, moebius(n/d)^2*d^6);
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^6*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Wesley Ivan Hurt, Feb 06 2022
STATUS
approved