For which $n$ does $n\mid 1^n+2^n+\cdots+n^n$? Find all the natural numbers $n$ such that
$$n\mid 1^n+2^n+\cdots+n^n.$$
We know through Faulhaber's formula, that 
$$\sum_{k=1}^{n}k^n=\frac1{n+1}\sum_{k=0}^n\binom{n+1}{k}B_k n^{n+1-k},$$
where $B_k$ is a Bernoulli number. I checked few dozen values of $n$ and it seems that only odd numbers are solutions. Any ideas on how to proceed from here?

To see tnat odd numbers always work, it is enough to use the formula $a^n+b^n = (a+b)(a^{n-1}-a^{n-2}b+...+b^n)$. Thus, if $n$ is odd, $n | k^n+(n-k)^n$ and therefore it divides the sum $1^n+...+n^n$.
