Integers that are not the sum of positive powers

The generalized Waring problem asks exactly which positive integers cannot be expressed as the sum of $j$ positive $k$-th powers? Using computational techniques, this paper refines an approach introduced by Zenkin, establishes results for the individual cases $5 \le k \le 9$, and resolves conjectures of Zenkin and the OEIS. This paper further establishes theoretical results regarding the properties of the sets of integers that are not the sum of $j$ positive $k$-th powers. The notion of Waring's problem is further extended to the finite sets of non-representable numbers where $G(1,k) < j < g(1,k)$. Improved computational techniques and results from Waring's problem are used throughout to catalog the sets of such integers, which are then considered in a general setting.