Asymptotic formula for M_k(d)

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

Background

The paper presents numerical values and curve-fitting experiments for M_k(d), including comparisons with polynomials in log d. The available computations do not determine the eventual asymptotic behavior.

The authors explicitly formulate the derivation of an asymptotic formula for each k as a problem, motivated by the apparent slow but nontrivial growth of M_k(d).

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 “Further Conjectures: Do kth Powers Repel Partition Numbers?”