This comes purely out of curiosity and experiments. I'm not sure if the literature has any pointers.

Let $p(n)$ be the number of integer partitions of $n$. Then, we have the well-known generating function $$\sum_{n\geq0}p(n)x^n=\prod_{k=1}^{\infty}\frac1{1-x^k}.$$

Question. For each fixed $k\in\mathbb{N}$, is the following set finite? $$\mathcal{A}_k:=\{(n,m)\in\mathbb{Z}_{\geq0}^2: p(n)+k=m^2\}.$$