A344524 - OEIS

%I #27 May 23 2021 07:16:30

%S 1,33,246,1060,3165,8091,17128,33936,60645,103825,164886,259368,

%T 381841,557595,784200,1091056,1462353,1968261,2554810,3327120,4230561,

%U 5361463,6644196,8302020,10113445,12352041,14873418,17924356,21225165,25341375,29670556,34920348,40625541,47297365

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

%C 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

%H Seiichi Manyama, <a href="/A344524/b344524.txt">Table of n, a(n) for n = 1..10000</a>

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

%F 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.

%F a(n) ~ Pi^4 * n^5 / (90*zeta(5)). - _Vaclav Kotesovec_, May 23 2021

%t a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^5, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, May 22 2021 *)

%o (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]))))));

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

%o (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))

%Y Column k=5 of A344479.

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

%K nonn

%O 1,2

%A _Seiichi Manyama_, May 22 2021