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

1,2

Comments

Dirichlet convolution of Euler totient function with tetrahedral 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) * (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

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}]

Crossrefs

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

Sequence in context: A038254 A223321 A073095 * A364970 A294017 A367379

Adjacent sequences: A309320 A309321 A309322 * A309324 A309325 A309326

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 23 2019

Status

approved