OFFSET
1,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^2.
a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_2(d), where mu is the Moebius function A008683.
From Amiram Eldar, May 21 2024: (Start)
Multiplicative with a(p^e) = p^(2*e-2) * (p^2 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-4)/zeta(s).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(3)/zeta(5) = 1.1592484598... . (End)
MATHEMATICA
f[p_, e_] := p^(2*e-2) * (p^2 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)
PROG
(PARI) a(n) = sumdiv(n, d, moebius(n/d)*d^2*sigma(d, 2));
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 17 2024
STATUS
approved