Repellent properties of perfect powers on partition functions: a heuristic approach

Abstract

In 2013, Sun conjectured that the partition function $p(n)$ is never a perfect power for $n\ge 2$. Building on this, Merca, Ono, and Tsai recently observed that for any fixed integers $d\ge 0$ and $k\ge 2$, there appear to be only finitely many integers $n$ such that $p(n)$ differs from a perfect $k$th power by at most $d$. Denoting by $M_k(d)$ the largest such $n$, they conjectured that $M_k(d)=o(d^{\epsilon })$ for every $\epsilon >0$. In this paper, we investigate the asymptotic growth of analogs of $M_k(d)$ for a wide class of partition functions. We establish sharp lower bounds and provide heuristics which suggest that $M_k(d)$ in fact grows polylogarithmically in $d$, i.e. of order ${\log }^2(d)$. More generally, we prove that if $f(n)$ is a suitably random chosen function with asymptotic growth rate similar to that of $p(n)$, then the set of integers $n$ for which $f(n)$ is a perfect power is finite with probability 1.