A372937 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^5.

1, 47, 323, 1744, 3749, 15181, 19207, 59648, 84969, 176203, 175691, 563312, 399853, 902729, 1210927, 1970176, 1503377, 3993543, 2606419, 6538256, 6203861, 8257477, 6716183, 19266304, 12105625, 18793091, 21172347, 33497008, 21218429, 56913569, 29552671, 64028672

Offset

1,2

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, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^4.

a(n) = Sum_{d|n} mu(n/d) * d^4 * sigma(d), where mu is the Moebius function A008683.

From Amiram Eldar, May 21 2024: (Start)

Multiplicative with a(p^e) = p^(4*e-4)*(p^e*(p^5-1) - (p^4-1))/(p-1).

Dirichlet g.f.: zeta(s-4)*zeta(s-5)/zeta(s).

Sum_{k=1..n} a(k) ~ c * n^6 / 6, c = zeta(2)/zeta(6) = 315/(2*Pi^4) = 1.616892... (A157292). (End)

Mathematica

f[p_, e_] := p^(4*e-4)*(p^e*(p^5-1) - (p^4-1))/(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^4*sigma(d));

Crossrefs

Cf. A343498, A372926, A372929, A372931.

Cf. A000203, A008683.

Cf. A013661, A013664, A157292.

Sequence in context: A140043 A074774 A107611 * A219442 A055828 A244770

Adjacent sequences: A372934 A372935 A372936 * A372938 A372939 A372940

Keyword

nonn,mult

Author

Seiichi Manyama, May 17 2024

Status

approved