# Must the sum of the digits of $n^k$ decrease infinitely often, for $n,k\in\mathbb{N}$ and $n$ not a power of $10$?

Let $S(n)$ be the sum of the digits of $n$ (in base $10$), is it true that for every $n$ which is not a power of $10$ there are infinitely many $k$ such that $S(n^k)>S(n^{k+1})$? (here $n$ and $k$ are clearly both integers)
The case $n=2$ is answered in this MSE question, what can be said about other values of $n$?
• A similar problem involving varying $n$ and fixed $k$ has been answered by this paper: arxiv.org/pdf/1311.0515v2.pdf Jul 11, 2015 at 12:38