A254521 Coefficients of the Dirichlet series zeta(s-3) / zeta(3s-3).

1, 8, 27, 64, 125, 216, 343, 504, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4032, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13608, 15625, 17576, 19656, 21952, 24389, 27000, 29791, 32256, 35937, 39304, 42875, 46656, 50653, 54872, 59319

Offset

1,2

Links

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Formula

a(n) = n^3 * Sum_{d^3 | n} (moebius(d) / d^6).

Multiplicative with a(p) = p^3; a(p^2) = p^6; a(p^e) = p^(3e) - p^(3e-6), for e > 2.

Sum_{k=1..n} a(k) ~ n^4 / (4*Zeta(9)). - Vaclav Kotesovec, Feb 03 2019

Sum_{k>=1} 1/a(k) = Product_{p prime} (1 + 1/p^3 + 1/p^6 + 1/((p^3 - 1)^2*(p^3 + 1))) = 1.202094253239358480267688474077353358147971390519883358936462981705245... - Vaclav Kotesovec, Sep 26 2020

Prog

(PARI) a(n) = n^3*sumdiv(n, d, if (ispower(d, 3), moebius(sqrtnint(d, 3))/d^2)); \\ Michel Marcus, Feb 10 2015
(PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1 - p^3*X)*(1 - p^3*X^3))[n], ", ")) \\ Vaclav Kotesovec, Sep 26 2020

Crossrefs

Cf. A000578.

Sequence in context: A076989 A270437 A259603 * A351985 A352172 A376270

Adjacent sequences: A254518 A254519 A254520 * A254522 A254523 A254524

Keyword

mult,nonn

Author

Álvar Ibeas, Jan 31 2015

Status

approved