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A344522 a(n) = Sum_{1 <= i, j, k <= n} gcd(i,j,k).

%I A344522 #30 May 23 2021 07:02:21
%S A344522 1,9,30,76,141,267,400,624,885,1249,1590,2208,2689,3411,4248,5248,
%T A344522 6081,7485,8530,10248,11889,13687,15228,17988,20053,22569,25242,28588,
%U A344522 31053,35463,38284,42540,46581,50893,55362,61824,65857,71247,76884,84388,89349,97881,103342
%N A344522 a(n) = Sum_{1 <= i, j, k <= n} gcd(i,j,k).
%H A344522 Seiichi Manyama, <a href="/A344522/b344522.txt">Table of n, a(n) for n = 1..10000</a>
%F A344522 a(n) = Sum_{k=1..n} phi(k) * floor(n/k)^3.
%F A344522 G.f.: (1/(1 - x)) * Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^3.
%F A344522 a(n) ~ Pi^2 * n^3 / (6*zeta(3)). - _Vaclav Kotesovec_, May 23 2021
%t A344522 a[n_] := Sum[EulerPhi[k] * Quotient[n, k]^3, {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, May 22 2021 *)
%o A344522 (PARI) a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, gcd([i, j, k]))));
%o A344522 (PARI) a(n) = sum(k=1, n, eulerphi(k)*(n\k)^3);
%o A344522 (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+4*x^k+x^(2*k))/(1-x^k)^3)/(1-x))
%Y A344522 Column k=3 of A344479.
%Y A344522 Cf. A018806, A071778, A343497, A344132, A344521, A344523, A344524, A344525, A344526.
%K A344522 nonn
%O A344522 1,2
%A A344522 _Seiichi Manyama_, May 22 2021