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A309322 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^3, where phi = Euler totient function (A000010).

%I A309322 #23 May 26 2025 11:47:46
%S A309322 1,4,8,15,19,35,34,56,63,86,76,141,103,157,182,212,169,294,208,355,
%T A309322 335,359,298,556,405,490,522,657,463,865,526,816,773,812,856,1239,739,
%U A309322 1003,1058,1424,901,1610,988,1525,1617,1445,1174,2188,1435,1960,1760,2091,1483,2529,1994
%N A309322 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^3, where phi = Euler totient function (A000010).
%C A309322 Dirichlet convolution of Euler totient function with triangular numbers.
%H A309322 Vaclav Kotesovec, <a href="/A309322/b309322.txt">Table of n, a(n) for n = 1..10000</a>
%F A309322 a(n) = Sum_{d|n} phi(n/d) * d * (d + 1)/2.
%F A309322 a(n) = Sum_{k=1..n} Sum_{j=1..k} gcd(j,k,n).
%F A309322 a(n) = Sum_{k=1..n} gcd(n,k)*(gcd(n,k)+1)/2. - _Richard L. Ollerton_, May 07 2021
%F A309322 Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (36*zeta(3)). - _Vaclav Kotesovec_, May 23 2021
%F A309322 a(n) = (A018804(n) + A069097(n))/2. - _Ridouane Oudra_, May 22 2025
%t A309322 nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A309322 Table[Sum[EulerPhi[n/d] d (d + 1)/2, {d, Divisors[n]}], {n, 1, 55}]
%t A309322 Table[Sum[Sum[GCD[j, k, n], {j, 1, k}], {k, 1, n}], {n, 1, 55}]
%Y A309322 Cf. A000010, A000217, A018804, A069097, A272718, A309323.
%K A309322 nonn
%O A309322 1,2
%A A309322 _Ilya Gutkovskiy_, Jul 23 2019