OFFSET
1,2
COMMENTS
Sum of the 8th 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^8 * mu(n/d)^2.
a(n) = n^8 * Sum_{d|n} mu(d)^2 / d^8.
Multiplicative with a(p^e) = p^(8*e) + p^(8*e-8). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^9 * zeta(9) / (9 * zeta(18)) = 4331032831125 * n^9 * zeta(9) / (43867 * Pi^18).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^8/(p^16-1)) = 1.004062071480173688638170669970682370243496458304179434830922739661777... (End)
a(n) = J_16(n)/J_8(n) = J_16(n)/A069093(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Nov 14 2022
MATHEMATICA
f[p_, e_] := p^(8*e) + p^(8*(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^8);
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^8*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Wesley Ivan Hurt, Feb 06 2022
STATUS
approved