HMMT 2020 A5

A positive integer $N$ is piquant if there exists a positive integer $m$ such that if $n_{i}$ denotes the number of digits in $m^{i}$ (in base 10), then $n_{1}+n_{2}+\cdots+n_{10}=N$. Let $p_{M}$ denote the fraction of the first $M$ positive integers that are piquant. Find $\lim _{M \rightarrow \infty} p_{M}$.