OFFSET
1,2
COMMENTS
Apparently, a(n) == 1 (mod 240). - Hugo Pfoertner, May 20 2024
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^4.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_8(d).
From Amiram Eldar, May 20 2024: (Start)
Multiplicative with a(p^e) = (p^(8*e + 4) + 1)/(p^4 + 1).
Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(s-4).
Sum_{k=1..n} a(k) ~ (zeta(9)/(9*zeta(5))) * n^9. (End)
MATHEMATICA
f[p_, e_] := (p^(8*e + 4) + 1)/(p^4 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, May 20 2024 *)
PROG
(PARI) a(n) = sigma(n^2, 8)/sigma(n^2, 4);
(PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 8));
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 18 2024
STATUS
approved