A372929 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^4.

1, 23, 107, 424, 749, 2461, 2743, 7232, 9369, 17227, 15971, 45368, 30757, 63089, 80143, 119296, 88433, 215487, 137179, 317576, 293501, 367333, 292007, 773824, 483625, 707411, 777843, 1163032, 731669, 1843289, 953311, 1937408, 1708897, 2033959, 2054507, 3972456

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 <= n} gcd(x_1, x_2, x_3, x_4, n)^3.

a(n) = Sum_{d|n} mu(n/d) * d^3 * sigma(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^(e+1)-1) - p^e + 1)/(p-1).

Dirichlet g.f.: zeta(s-3)*zeta(s-4)/zeta(s).

Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2)/zeta(5) = 1.586353589... . (End)

Mathematica

f[p_, e_] := p^(3*e-3) * (p^3 * (p^(e+1)-1) - p^e + 1)/(p-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));

Crossrefs

Cf. A343497, A368743, A372928, A372930.

Cf. A343498, A372926, A372931.

Cf. A000203, A008683.

Cf. A013661, A013663.

Sequence in context: A180517 A142244 A139912 * A176928 A157607 A142272

Adjacent sequences: A372926 A372927 A372928 * A372930 A372931 A372932

Keyword

nonn,mult

Author

Seiichi Manyama, May 17 2024

Status

approved