OFFSET
1,2
COMMENTS
Sum of the 7th 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^7 * mu(n/d)^2.
a(n) = n^7 * Sum_{d|n} mu(d)^2 / d^7.
Multiplicative with a(p^e) = p^(7*e) + p^(7*e-7). - Sebastian Karlsson, Feb 08 2022
From Vaclav Kotesovec, Feb 12 2022: (Start)
Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^8 * zeta(8) / (8 * zeta(16)) = 34459425 * n^8 / (28936 * Pi^8).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^7/(p^14-1)) = 1.008287998838997802253937842472728682107868602338715231926150271159410... (End)
a(n) = J_14(n) / J_7(n) = J_14(n) / A069092(n), where J_k is the k-th Jordan totient function. - Enrique Pérez Herrero, Nov 13 2022
MATHEMATICA
f[p_, e_] := p^(7*e) + p^(7*(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^7);
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^7*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Wesley Ivan Hurt, Feb 06 2022
STATUS
approved