A344524 a(n) = Sum_{1 <= i, j, k, l, m <= n} gcd(i,j,k,l,m).

1, 33, 246, 1060, 3165, 8091, 17128, 33936, 60645, 103825, 164886, 259368, 381841, 557595, 784200, 1091056, 1462353, 1968261, 2554810, 3327120, 4230561, 5361463, 6644196, 8302020, 10113445, 12352041, 14873418, 17924356, 21225165, 25341375, 29670556, 34920348, 40625541, 47297365

Offset: 1

Author

Seiichi Manyama, May 22 2021

Keywords

• nonn

Comments

In general, for m > 2, Sum_{k=1..n} phi(k) * floor(n/k)^m ~ zeta(m-1) * n^m / zeta(m). - Vaclav Kotesovec, May 23 2021

Links

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

Crossrefs

Column k=5 of A344479.

Cf. A082544, A343499, A344139, A344522, A344523, A344525.

Programs

Mathematica

a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^5, {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, sum(m=1, n, gcd([i, j, k, l, m]))))));

PARI

a(n) = sum(k=1, n, eulerphi(k)*(n\k)^5);

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+26*x^k+66*x^(2*k)+26*x^(3*k)+x^(4*k))/(1-x^k)^5)/(1-x))

Formula

a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^5.

G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^5.

a(n) ~ Pi^4 * n^5 / (90*zeta(5)). - Vaclav Kotesovec, May 23 2021