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

1,2

Comments

Dirichlet convolution of Euler totient function with triangular numbers.

Links

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

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

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}]

Crossrefs

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

Sequence in context: A267368 A345096 A190692 * A312744 A312745 A312746

Adjacent sequences: A309319 A309320 A309321 * A309323 A309324 A309325

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 23 2019

Status

approved