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

1, 5, 12, 26, 39, 76, 90, 152, 191, 275, 296, 492, 467, 674, 798, 1000, 985, 1467, 1348, 1934, 2011, 2360, 2322, 3420, 3085, 3791, 4062, 4944, 4523, 6454, 5486, 7168, 7237, 8189, 8340, 10942, 9175, 11300, 11714, 14208, 12381, 16759, 14232, 18036, 18549, 19706, 18470

Offset: 1

Author

Ilya Gutkovskiy, Jul 23 2019

Keywords

• nonn

Comments

Dirichlet convolution of Euler totient function with tetrahedral numbers.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

nmax = 47; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[EulerPhi[n/d] d (d + 1) (d + 2)/6, {d, Divisors[n]}], {n, 1, 47}]
Table[Sum[Sum[Sum[GCD[i, j, k, n], {i, 1, j}], {j, 1, k}], {k, 1, n}], {n, 1, 47}]

Formula

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

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

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