A344521 a(n) = Sum_{1 <= i <= j <= k <= n} gcd(i,j,k).

1, 5, 13, 28, 47, 82, 116, 172, 235, 321, 397, 538, 641, 798, 980, 1192, 1361, 1655, 1863, 2218, 2553, 2912, 3210, 3766, 4171, 4661, 5183, 5840, 6303, 7168, 7694, 8510, 9283, 10095, 10951, 12190, 12929, 13932, 14990, 16414, 17315, 18925, 19913, 21438, 23055, 24500, 25674, 27862

Offset: 1

Author

Seiichi Manyama, May 22 2021

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Column k=3 of A345229.

Partial sums of A309322.

Cf. A272718, A344132, A344522, A344992.

Programs

Mathematica

a[n_] := Sum[Sum[Sum[GCD[i, j, k], {i, 1, j}], {j, 1, k}], {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 25 2021 *)
nmax = 100; Rest[CoefficientList[Series[1/(1 - x)*Sum[EulerPhi[k]*x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 05 2021 *)
Accumulate[Table[Sum[EulerPhi[n/d] * d*(d+1)/2, {d, Divisors[n]}], {n, 1, 100}]] (* Vaclav Kotesovec, Jun 05 2021 *)

PARI

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

Formula

From Vaclav Kotesovec, Jun 05 2021: (Start)

a(n) ~ Pi^2 * n^3 / (36*zeta(3)).

G.f.: 1/(1-x) * Sum_{k>=1} phi(k) * x^k/(1 - x^k)^3, where phi is the Euler totient function (A000010).

a(n) = Sum_{k=1..n} Sum_{d|k} phi(k/d) * d*(d+1)/2. (End)

a(n) = Sum_{k=1..n} phi(k) * binomial(floor(n/k)+2,3). - Seiichi Manyama, Sep 13 2024