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a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i,j,k,l).
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%I #26 May 23 2021 07:08:00

%S 1,17,84,276,649,1417,2528,4432,7033,10905,15556,22836,30673,41729,

%T 54944,71968,89969,115457,140820,175444,212537,257113,302720,366160,

%U 426505,500873,580676,677108,769761,895377,1008928,1153120,1300417,1469073,1640020,1860340,2054921

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

%H Seiichi Manyama, <a href="/A344523/b344523.txt">Table of n, a(n) for n = 1..10000</a>

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

%F 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.

%F a(n) ~ 90 * zeta(3) * n^4 / Pi^4. - _Vaclav Kotesovec_, May 23 2021

%t a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^4, {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, gcd([i, j, k, l])))));

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

%o (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))

%Y Column k=4 of A344479.

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

%K nonn

%O 1,2

%A _Seiichi Manyama_, May 22 2021