How would one find the maximal $n$ such that there exists an $n$-subset $S$ of $\mathbb{Z}^+$ such that $\forall a\subseteq S, \sum_{a\in A}a$ is either a perfect square or a perfect cube, or can one go arbitrarily large. This is the case if one loosens the restrictions to any perfect power. Also, what about the slightly more general case for perfect squares, cubes, $...$ up to $k$th powers for some $k$.

How would one find the maximal $n$ such that there exists an $n$-subset $S$ of $\mathbb{Z}^+$ such that $\forall A\subseteq S, \sum_{a\in A}a$ is either a perfect square or a perfect cube, or can one go arbitrarily large. This is the case if one loosens the restrictions to any perfect power. Also, what about the slightly more general case for perfect squares, cubes, $...$ up to $k$th powers for some $k$.