Let $n\ge 1$ be a positive integer and $S$ be the set of all $n$-tuples of positive integers $1\le a_1\le a_2\le \cdots \le a_n$ such that $$\sum_{j=1}^n a_j!$$ is a non-trivial power (a number $a^b$ with integers $a,b\ge 2$).

Since the factorial grows very fast, I conjecture that for every $n\ge 1$, there are only a finite many such tuples.

Can we prove that $S$ is finite for every $n\ge 1$ ?


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