A372952 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} n/gcd(x_1, x_2, x_3, n).

1, 15, 79, 239, 621, 1185, 2395, 3823, 6397, 9315, 14631, 18881, 28549, 35925, 49059, 61167, 83505, 95955, 130303, 148419, 189205, 219465, 279819, 302017, 388121, 428235, 518155, 572405, 707253, 735885, 923491, 978671, 1155849, 1252575, 1487295, 1528883

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_4(d).

From Amiram Eldar, May 21 2024: (Start)

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

Dirichlet g.f.: zeta(s)*zeta(s-4)/zeta(s-1).

Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(5)/zeta(4) = 0.958057374... . (End)

a(n) = Sum_{d|n} phi(n/d) * (n/d) * sigma_4(d^2)/sigma_2(d^2). - Seiichi Manyama, May 24 2024

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) )^3. - Seiichi Manyama, May 25 2024

Mathematica

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

Crossrefs

Column k=3 of A372968.

Cf. A001159, A008683.

Cf. A013662, A013663.

Cf. A371492, A372962.

Sequence in context: A212746 A212741 A082540 * A269657 A189922 A085808

Adjacent sequences: A372949 A372950 A372951 * A372953 A372954 A372955

Keyword

nonn,mult

Author

Seiichi Manyama, May 18 2024

Status

approved