Sun's perfect-power repulsion conjecture for the partition function

Let $p(n)$ denote the number of unrestricted integer partitions of $n$, with $p(0)=1$. An integer is a perfect power if it has the form $m^k$ with integers $m\ge 2$ and $k\ge2$. For fixed $k\ge2$, define the distance from $p(n)$ to the nearest $k$th power by

$$\Delta_k(n):=\min_{m\ge0}|p(n)-m^k|.$$

Sun's perfect-power repulsion conjecture. The partition function $p(n)$ repels perfect powers in the following senses: (i) for every $n\ge2$ and $k\ge2$, $p(n)\neq m^k$ for all integers $m\ge2$; and (ii) for every $k\ge2$ and $d\ge0$, only finitely many $n$ satisfy $\Delta_k(n)\le d$.

The first assertion is Sun's original no-perfect-powers conjecture, while the second is the stronger perfect-power repulsion conjecture developed with later generalizations by Merca et al. The paper supports these conjectures through analogous finiteness results for restricted partition functions $p_B(n)$, rather than proving the stated claims for $p(n)$ itself.