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A309323 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^4, where phi = Euler totient function (A000010).

%I A309323 #9 May 23 2021 08:35:44
%S A309323 1,5,12,26,39,76,90,152,191,275,296,492,467,674,798,1000,985,1467,
%T A309323 1348,1934,2011,2360,2322,3420,3085,3791,4062,4944,4523,6454,5486,
%U A309323 7168,7237,8189,8340,10942,9175,11300,11714,14208,12381,16759,14232,18036,18549,19706,18470
%N A309323 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^4, where phi = Euler totient function (A000010).
%C A309323 Dirichlet convolution of Euler totient function with tetrahedral numbers.
%H A309323 Vaclav Kotesovec, <a href="/A309323/b309323.txt">Table of n, a(n) for n = 1..10000</a>
%F A309323 a(n) = Sum_{d|n} phi(n/d) * d * (d + 1) * (d + 2)/6.
%F A309323 a(n) = Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j} gcd(i,j,k,n).
%F A309323 Sum_{k=1..n} a(k) ~ 15 * zeta(3) * n^4 / (4*Pi^4). - _Vaclav Kotesovec_, May 23 2021
%t A309323 nmax = 47; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A309323 Table[Sum[EulerPhi[n/d] d (d + 1) (d + 2)/6, {d, Divisors[n]}], {n, 1, 47}]
%t A309323 Table[Sum[Sum[Sum[GCD[i, j, k, n], {i, 1, j}], {j, 1, k}], {k, 1, n}], {n, 1, 47}]
%Y A309323 Cf. A000010, A000292, A018804, A272718, A309322.
%K A309323 nonn
%O A309323 1,2
%A A309323 _Ilya Gutkovskiy_, Jul 23 2019