My friend said that for any $n=p^a$, where $p$ is odd prime, $a$ is positive integer then: If $k$ is divisible by $p-1$ then $1^k+2^k+\cdots+n^k\equiv -p^{a-1}\pmod{p^a}$. I am very sure that his result is wrong. My though is simple: I use primitive root of $p^a$. But I fail to construct a counterexample for him. So my question is that: Could we construct an example of $k$ so that $p-1\mid k$ and $1^k+2^k+\cdots+n^k\not\equiv -p^{a-1}\pmod{p^a}$.
The second question is that: Does the following holds: $1^k+2^k+\cdots+n^k\not\equiv -p^{a-1}\pmod{p^a}$ if and only if $p-1\mid k$?
