A373059 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, n)/gcd(x_1, x_2, n).

1, 5, 13, 25, 41, 65, 85, 121, 157, 205, 221, 325, 313, 425, 533, 569, 545, 785, 685, 1025, 1105, 1105, 1013, 1573, 1441, 1565, 1777, 2125, 1625, 2665, 1861, 2617, 2873, 2725, 3485, 3925, 2665, 3425, 4069, 4961, 3281, 5525, 3613, 5525, 6437, 5065, 4325, 7397, 5965

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Formula

a(n) = Sum_{d|n} phi(n/d) * (n/d) * sigma_2(d^2)/sigma(d^2).

From Amiram Eldar, May 27 2024: (Start)

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

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

Sum_{k=1..n} a(k) ~ (2*zeta(3)*n^3/(15*zeta(4))) * (log(n) + 2*gamma - 1/3 - 2*zeta'(2)/zeta(2) + zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)

Mathematica

f[p_, e_] := (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 27 2024 *)

Prog

(PARI) a(n) = sum(i=1, n, sum(j=1, n, gcd(i, n)/gcd([i, j, n])));
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2); } \\ Amiram Eldar, May 27 2024

Crossrefs

Row sums of A350900.

Cf. A057660, A350156.

Cf. A001620, A002117, A013661, A073002, A244115, A306016.

Sequence in context: A133322 A299258 A301671 * A268525 A146590 A098483

Adjacent sequences: A373056 A373057 A373058 * A373060 A373061 A373062

Keyword

nonn,mult

Author

Seiichi Manyama, May 21 2024

Status

approved