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

1, 4, 8, 15, 19, 35, 34, 56, 63, 86, 76, 141, 103, 157, 182, 212, 169, 294, 208, 355, 335, 359, 298, 556, 405, 490, 522, 657, 463, 865, 526, 816, 773, 812, 856, 1239, 739, 1003, 1058, 1424, 901, 1610, 988, 1525, 1617, 1445, 1174, 2188, 1435, 1960, 1760, 2091, 1483, 2529, 1994

Offset: 1

Author

Ilya Gutkovskiy, Jul 23 2019

Keywords

• nonn

Comments

Dirichlet convolution of Euler totient function with triangular numbers.

Links

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

Crossrefs

Cf. A000010, A000217, A018804, A069097, A272718, A309323.

Programs

Mathematica

nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[EulerPhi[n/d] d (d + 1)/2, {d, Divisors[n]}], {n, 1, 55}]
Table[Sum[Sum[GCD[j, k, n], {j, 1, k}], {k, 1, n}], {n, 1, 55}]

Formula

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

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

a(n) = Sum_{k=1..n} gcd(n,k)*(gcd(n,k)+1)/2. - Richard L. Ollerton, May 07 2021

Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (36*zeta(3)). - Vaclav Kotesovec, May 23 2021

a(n) = (A018804(n) + A069097(n))/2. - Ridouane Oudra, May 22 2025