If $n\geq 2$ is prime and $a\geq 1$ is an integer, then $(a+1)^n-a^n$ leaves a remainder of $1$ modulo $p$.
If $n$ is even or a multiple of $3$, then it follows ad-hoc that again $(a+1)^n-a^n$ is never divisible by $n$.
By simulations, the same should be true for all $n$. How would one prove this?
Prove that $(a+1)^n-a^n$ is never divisible by $n$.