A372930 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^5.

1, 39, 269, 1304, 3249, 10491, 17149, 42176, 66069, 126711, 162381, 350776, 373489, 668811, 873981, 1353216, 1424769, 2576691, 2482957, 4236696, 4613081, 6332859, 6448509, 11345344, 10168625, 14566071, 16073721, 22362296, 20535537, 34085259, 28658941, 43331584

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^3.

a(n) = Sum_{d|n} mu(n/d) * d^3 * sigma_2(d), where mu is the Moebius function A008683.

From Amiram Eldar, May 21 2024: (Start)

Multiplicative with a(p^e) = p^(3*e-3) * (p^3 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1).

Dirichlet g.f.: zeta(s-3)*zeta(s-5)/zeta(s).

Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(3)/zeta(6) = 1.181564... (A157289). (End)

Mathematica

f[p_, e_] := p^(3*e-3) * (p^3 * (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^3*sigma(d, 2));

Crossrefs

Cf. A343497, A368743, A372928, A372929.

Cf. A001157, A008683.

Cf. A002117, A013664, A157289.

Sequence in context: A101627 A229639 A070146 * A327344 A190537 A187095

Adjacent sequences: A372927 A372928 A372929 * A372931 A372932 A372933

Keyword

nonn,mult

Author

Seiichi Manyama, May 17 2024

Status

approved