A344523 a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i,j,k,l).

1, 17, 84, 276, 649, 1417, 2528, 4432, 7033, 10905, 15556, 22836, 30673, 41729, 54944, 71968, 89969, 115457, 140820, 175444, 212537, 257113, 302720, 366160, 426505, 500873, 580676, 677108, 769761, 895377, 1008928, 1153120, 1300417, 1469073, 1640020, 1860340, 2054921

Offset: 1

Author

Seiichi Manyama, May 22 2021

Keywords

• nonn

Links

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

Crossrefs

Column k=4 of A344479.

Cf. A082540, A343498, A344138, A344522, A344524, A344525.

Programs

Mathematica

a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^4, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 22 2021 *)

PARI

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

PARI

a(n) = sum(k=1, n, eulerphi(k)*(n\k)^4);

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^4)/(1-x))

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^4.

G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^4.

a(n) ~ 90 * zeta(3) * n^4 / Pi^4. - Vaclav Kotesovec, May 23 2021