A373061 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, n).

1, 9, 31, 77, 141, 279, 379, 637, 877, 1269, 1431, 2387, 2341, 3411, 4371, 5181, 5169, 7893, 7183, 10857, 11749, 12879, 12651, 19747, 18041, 21069, 24043, 29183, 25173, 39339, 30691, 41789, 44361, 46521, 53439, 67529, 51949, 64647, 72571, 89817, 70521, 105741

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from Seiichi Manyama)

Formula

From Amiram Eldar, May 31 2024: (Start)

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

Dirichlet g.f.: zeta(s)*zeta(s-2)*zeta(s-3)/zeta(s-1)^2.

Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(2) * zeta(4) / zeta(3)^2 = Pi^6/(540*zeta(3)^2) = 1.232126852811... . (End)

Mathematica

f[p_, e_] := (p^(3*e)*(p+1)^3 - p^(2*e)*(p^2+p+1) + 1)/((p^2+p+1)*(p+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 31 2024 *)

Prog

(PARI) a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, gcd([i, j, n])/gcd([i, j, k, n]))));

Crossrefs

Cf. A372952, A373060.

Cf. A002117, A013661, A013662.

Sequence in context: A344675 A177342 A224000 * A118444 A321598 A048374

Adjacent sequences: A373058 A373059 A373060 * A373062 A373063 A373064

Keyword

nonn,mult

Author

Seiichi Manyama, May 21 2024

Status

approved