A230283 Numerators to Dirichlet inverse of Euler totient based version of series expansion for x/LambertW(x).

1, 1, -1, 2, -9, 8, -625, 2, -117649, 128, -6561, 8, -25937424601, 18, -23298085122481, 16, -9, 32768, -48661191875666868481, 400, -104127350297911241532841, 648, -81, 256, -907846434775996175406740561329, 490, -59604644775390625, 1024, -2541865828329, 1296

Offset

1,4

Comments

The coefficients of the series expansion of x/Lambert(x) expanded at 0 can be seen as exponentiated numerators in convergents of zeta function limits of truncated Dirichlet series for logarithms. Those numerators are defined by simple recurrences. Letting those recurrences run in cross directions to each other, one get the Dirichlet inverse of the Euler totient in a greatest common divisor matrix, and the von Mangoldt function as convergents of Dirichlet series. Since x/LambertW(x) is good at approximately describing the nontrivial Riemann zeta zeros and since the Riemann zeta zeros are the frequencies that build up the von Mangoldt function, this prime number or von Mangoldt function version of the x/LambertW(x) is motivated.

Links

G. C. Greubel, Table of n, a(n) for n = 1..385

Mathematica

Clear[nn, n, k, s, x]; nn = 22; Numerator[CoefficientList[1 + Integrate[1 + Expand[Sum[Exp[Limit[Zeta[s]*Sum[(If[n == 1, 0, Table[DivisorSum[m, # MoebiusMu[#] &], {m, nn}][[GCD[n, k]]]])/(k)^(s - 1), {k, 1, n}], s -> 1]]*(-x)^n, {n, 1, nn}]], x], x]]

Crossrefs

Cf. A191898, A177885, A230284 (denominators).

Sequence in context: A249225 A191351 A324553 * A121067 A073904 A036879

Adjacent sequences: A230280 A230281 A230282 * A230284 A230285 A230286

Keyword

sign,frac

Author

Mats Granvik, Oct 15 2013

Status

approved