Waring numbers of ramified $p$-adic rings

Let $p$ be prime, $e,k$ be positive integers, and let $R = \mathbb{Z}_p[\sqrt[e]{p}]$. We calculate the Waring numbers $g_R(k)$ for many values of $p, e$, and $k$, and investigate how the Waring numbers for $p=2$ change as $e$ and $k$ vary.