A372928 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^3.

1, 15, 53, 176, 249, 795, 685, 1856, 2133, 3735, 2661, 9328, 4393, 10275, 13197, 18432, 9825, 31995, 13717, 43824, 36305, 39915, 24333, 98368, 46625, 65895, 76545, 120560, 48777, 197955, 59581, 176128, 141033, 147375, 170565, 375408, 101305, 205755, 232829, 462144

Offset: 1

Author

Seiichi Manyama, May 17 2024

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Column k=3 of A372938.

Cf. A343497, A368743, A372929, A372930.

Cf. A000005, A008683.

Cf. A001620, A013662, A261506.

Programs

Mathematica

f[p_, e_] := (e - e/p^3 + 1) * p^(3*e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

PARI

a(n) = sumdiv(n, d, moebius(n/d)*d^3*numdiv(d));

Formula

a(n) = Sum_{d|n} mu(n/d) * d^3 * tau(d), where mu is the Moebius function A008683.

From Amiram Eldar, May 21 2024: (Start)

Multiplicative with a(p^e) = (e - e/p^3 + 1) * p^(3*e).

Dirichlet g.f.: zeta(s-3)^2/zeta(s).

Sum_{k=1..n} a(k) ~ (n^4/(4*zeta(4))) * (log(n) + 2*gamma - 1/4 - zeta'(4)/zeta(4)), where gamma is Euler's constant (A001620). (End)