A309323 - OEIS

%I #9 May 23 2021 08:35:44

%S 1,5,12,26,39,76,90,152,191,275,296,492,467,674,798,1000,985,1467,

%T 1348,1934,2011,2360,2322,3420,3085,3791,4062,4944,4523,6454,5486,

%U 7168,7237,8189,8340,10942,9175,11300,11714,14208,12381,16759,14232,18036,18549,19706,18470

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

%C Dirichlet convolution of Euler totient function with tetrahedral numbers.

%H Vaclav Kotesovec, <a href="/A309323/b309323.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} phi(n/d) * d * (d + 1) * (d + 2)/6.

%F a(n) = Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j} gcd(i,j,k,n).

%F Sum_{k=1..n} a(k) ~ 15 * zeta(3) * n^4 / (4*Pi^4). - _Vaclav Kotesovec_, May 23 2021

%t nmax = 47; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Table[Sum[EulerPhi[n/d] d (d + 1) (d + 2)/6, {d, Divisors[n]}], {n, 1, 47}]

%t Table[Sum[Sum[Sum[GCD[i, j, k, n], {i, 1, j}], {j, 1, k}], {k, 1, n}], {n, 1, 47}]

%Y Cf. A000010, A000292, A018804, A272718, A309322.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Jul 23 2019