A372926 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^4.

1, 19, 89, 316, 649, 1691, 2449, 5104, 7281, 12331, 14761, 28124, 28729, 46531, 57761, 81856, 83809, 138339, 130681, 205084, 217961, 280459, 280369, 454256, 406225, 545851, 590409, 773884, 708121, 1097459, 924481, 1310464, 1313729, 1592371, 1589401, 2300796

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)^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

Cf. A069097, A360428, A368743, A372927.

Cf. A343498, A372929, A372931.

Cf. A001157, A008683.

Cf. A002117, A013663.

Sequence in context: A039454 A142089 A174819 * A124947 A126406 A201306

Adjacent sequences: A372923 A372924 A372925 * A372927 A372928 A372929

Keyword

nonn,mult

Author

Seiichi Manyama, May 17 2024

Status

approved