A344522 a(n) = Sum_{1 <= i, j, k <= n} gcd(i,j,k).

1, 9, 30, 76, 141, 267, 400, 624, 885, 1249, 1590, 2208, 2689, 3411, 4248, 5248, 6081, 7485, 8530, 10248, 11889, 13687, 15228, 17988, 20053, 22569, 25242, 28588, 31053, 35463, 38284, 42540, 46581, 50893, 55362, 61824, 65857, 71247, 76884, 84388, 89349, 97881, 103342

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^3.

G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^3.

a(n) ~ Pi^2 * n^3 / (6*zeta(3)). - Vaclav Kotesovec, May 23 2021

Mathematica

a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)

Prog

(PARI) a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, gcd([i, j, k]))));
(PARI) a(n) = sum(k=1, n, eulerphi(k)*(n\k)^3);
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^3)/(1-x))

Crossrefs

Column k=3 of A344479.

Cf. A018806, A071778, A343497, A344132, A344521, A344523, A344524, A344525, A344526.

Sequence in context: A002414 A273604 A273640 * A291919 A301988 A334853

Adjacent sequences: A344519 A344520 A344521 * A344523 A344524 A344525

Keyword

nonn

Author

Seiichi Manyama, May 22 2021

Status

approved