OFFSET
1,2
COMMENTS
Sum of the 5th 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^5 * mu(n/d)^2.
a(n) = n^5 * Sum_{d|n} mu(d)^2 / d^5.
Multiplicative with a(p^e) = p^(5*e) + p^(5*e-5). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^6 * zeta(6) / (6 * zeta(12)) = 225225 * n^6 / (1382 * Pi^6).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^5/(p^10-1)) = 1.03592823428850098309076014982275428113698561633329794485946580153004... (End)
a(n) = J_10(n) / J_5(n) = A069095(n) / A059378(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Nov 13 2022
MATHEMATICA
f[p_, e_] := p^(5*e) + p^(5*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Feb 08 2022 *)
PROG
(PARI) a(n)=sumdiv(n, d, moebius(n/d)^2*d^5);
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^5*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Wesley Ivan Hurt, Feb 06 2022
STATUS
approved