elementary number theory - Does there exist an $a$ such that $a^n+1$ is divisible by $n^3$ for infinitely many $n$?

It is well known that there are infinitely many positive integers $n$ such that $2^n+1$ is divisible by $n$.

Also it is well known that there exist infinitely many positive integers $n$ such that $4^n+1$ is divisible by $n^2$.

But I still cannot find any positive integer $a$ for which there exist infinitely many positive integers $n$ such that $a^n+1$ (or $a^n-1$) is divisible by $n^3$.

How can I find such and $a$ or prove that it doesn't exist?