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A344521 a(n) = Sum_{1 <= i <= j <= k <= n} gcd(i,j,k).

%I A344521 #30 Sep 13 2024 11:59:53
%S A344521 1,5,13,28,47,82,116,172,235,321,397,538,641,798,980,1192,1361,1655,
%T A344521 1863,2218,2553,2912,3210,3766,4171,4661,5183,5840,6303,7168,7694,
%U A344521 8510,9283,10095,10951,12190,12929,13932,14990,16414,17315,18925,19913,21438,23055,24500,25674,27862
%N A344521 a(n) = Sum_{1 <= i <= j <= k <= n} gcd(i,j,k).
%H A344521 Vaclav Kotesovec, <a href="/A344521/b344521.txt">Table of n, a(n) for n = 1..10000</a>
%F A344521 From _Vaclav Kotesovec_, Jun 05 2021: (Start)
%F A344521 a(n) ~ Pi^2 * n^3 / (36*zeta(3)).
%F A344521 G.f.: 1/(1-x) * Sum_{k>=1} phi(k) * x^k/(1 - x^k)^3, where phi is the Euler totient function (A000010).
%F A344521 a(n) = Sum_{k=1..n} Sum_{d|k} phi(k/d) * d*(d+1)/2. (End)
%F A344521 a(n) = Sum_{k=1..n} phi(k) * binomial(floor(n/k)+2,3). - _Seiichi Manyama_, Sep 13 2024
%t A344521 a[n_] := Sum[Sum[Sum[GCD[i, j, k], {i, 1, j}], {j, 1, k}], {k, 1, n}]; Array[a, 50] (* _Amiram Eldar_, May 25 2021 *)
%t A344521 nmax = 100; Rest[CoefficientList[Series[1/(1 - x)*Sum[EulerPhi[k]*x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jun 05 2021 *)
%t A344521 Accumulate[Table[Sum[EulerPhi[n/d] * d*(d+1)/2, {d, Divisors[n]}], {n, 1, 100}]] (* _Vaclav Kotesovec_, Jun 05 2021 *)
%o A344521 (PARI) a(n) = sum(i=1, n, sum(j=i, n, sum(k=j, n, gcd([i, j, k]))));
%Y A344521 Column k=3 of A345229.
%Y A344521 Partial sums of A309322.
%Y A344521 Cf. A272718, A344132, A344522, A344992.
%K A344521 nonn
%O A344521 1,2
%A A344521 _Seiichi Manyama_, May 22 2021