 Solution

Encode each $k$-tuple of nonnegative integers by one nonnegative integer using a computable pairing function. Define a unary partial procedure $P$ on input $m$ as follows: use dovetailing on the computations of $f_{n,k}$ over all encoded $k$-tuples, and halt as soon as one of them halts with output $m$. ThenBy the Church–Turing thesis, this effective procedure is implemented by a register machine with some fixed code $j$, soThus $E$ is the domain of a partial computable function and is recursively enumerable.