Do perfect powers repel partition numbers?

TL;DR

The paper investigates the conjecture that partition numbers rarely or never are perfect powers, providing partial confirmations and proposing that powers tend to repel partition numbers, with several related conjectures about their distribution.

Contribution

It confirms Sun's conjecture in many cases and introduces new conjectures about the rarity and distribution of perfect powers near partition numbers.

Findings

Confirmed that partition numbers are rarely perfect powers in many cases.

Proposed that integral powers tend to repel partition numbers, with finitely many exceptions.

Conjectured bounds on the proximity of partition numbers to perfect powers.

Abstract

In 2013 Zhi-Wei Sun conjectured that $p(n)$ is never a power of an integer when $n>1.$ We confirm this claim in many cases. We also observe that integral powers appear to repel the partition numbers. If $k>1$ and $\Delta_k(n)$ is the distance between $p(n)$ and the nearest $k$th power, then for every $d\geq 0$ we conjecture that there are at most finitely many $n$ for which $\Delta_k(n)\leq d.$ More precisely, for every $\varepsilon>0,$ we conjecture that $$M_k(d):=\max\{n \ : \ \Delta_k(n)\leq d\}=o( d^{\varepsilon}).$$ In $k$-power aspect with $d$ fixed, we also conjecture that if $k$ is sufficiently large, then $$ M_k(d)=\max \left\{ n \ : \ p(n)-1\leq d\right\}. $$ In other words, $1$ generally appears to be the closest $k$th power among the partition numbers.