Does there exist $k \in N$ such that $ \sigma_k(n) $ is an injective function, where $\sigma_k(n)$ is the sum of the divisors of $n$ raised to the $k$th power. If so what is the minimum value of $k$ with this property?
The cases $k=1$ and $k=2$ are easily seen to be false, but for $k>2$, is there a known solution?