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

%I A344523 #26 May 23 2021 07:08:00
%S A344523 1,17,84,276,649,1417,2528,4432,7033,10905,15556,22836,30673,41729,
%T A344523 54944,71968,89969,115457,140820,175444,212537,257113,302720,366160,
%U A344523 426505,500873,580676,677108,769761,895377,1008928,1153120,1300417,1469073,1640020,1860340,2054921
%N A344523 a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i,j,k,l).
%H A344523 Seiichi Manyama, <a href="/A344523/b344523.txt">Table of n, a(n) for n = 1..10000</a>
%F A344523 a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^4.
%F A344523 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 A344523 a(n) ~ 90 * zeta(3) * n^4 / Pi^4. - _Vaclav Kotesovec_, May 23 2021
%t A344523 a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^4, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, May 22 2021 *)
%o A344523 (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 A344523 (PARI) a(n) = sum(k=1, n, eulerphi(k)*(n\k)^4);
%o A344523 (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 A344523 Column k=4 of A344479.
%Y A344523 Cf. A082540, A343498, A344138, A344522, A344524, A344525.
%K A344523 nonn
%O A344523 1,2
%A A344523 _Seiichi Manyama_, May 22 2021