cp's OEIS Frontend

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.

If prefixed by an additional 1, denominators of coefficients of expansion of exp(W(x)). - N. J. A. Sloane, Jan 08 2021

The functional equation formula in the answer by Peter Humphries is for the Dirichlet eta function and corresponds to the second term in this sequence. This sequence corresponds to zeta function products over all the divisors.