Let $\Bbb{Z}^+$be the set of all non-negative integers where $n$ and $k$ are given natural numbers. We consider the following non-decreasing function,
$$f:\Bbb{Z}^+ \to \Bbb{Z}^+$$ such that
\begin{align} f\left(\sum_{i=1}^{n} a_{i}^{n}\right) =\frac{1}{k}\sum_{i=1}^{n} \left(f(a_{i})\right)^{n}\\ \end{align}
1.Find all functions for $n=2014$ and $k=2014^{2013}$
2.Find, number of functions which satisfy the condition of the problem depending on the values of parameters $n$ and $k$.