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A309323 Expansion of Sum_{k>=1} phi(k) * x^k/(1 - x^k)^4, where phi = Euler totient function (A000010). 5
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 (list; graph; refs; listen; history; text; internal format)
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 * A294017 A212561 A199771

Adjacent sequences:  A309320 A309321 A309322 * A309324 A309325 A309326

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Jul 23 2019

STATUS

approved

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Last modified July 28 10:29 EDT 2021. Contains 346326 sequences. (Running on oeis4.)