Asymptotic formula for near-perfect-power indices

Determine, for each integer k > 1, an asymptotic formula for M_k(d) as d tends to infinity.

Background

The paper compares numerical values of M_k(d) with powers of log d and discusses whether these values might be asymptotically polynomial in log d. The available numerical fits are unstable and do not determine the eventual growth law.

The authors therefore pose the problem of finding an asymptotic formula for M_k(d) for each fixed exponent k.

References

It would be very interesting to carry out even further numerics to address the following problem. Problem. For each k, conjecture an asymptotic formula for the M_k(d).

— Do perfect powers repel partition numbers?  (2501.03754 - Merca et al., 7 Jan 2025) in Section 2, immediately before the concluding conjecture on the sequences M_k(d)