Settling some sum suppositions

We solve multiple conjectures by Byszewski and Ulas about the sum of base $b$ digits function. In order to do this, we develop general results about summations over the sum of digits function. As a corollary, we describe an unexpected new result about the Prouhet-Tarry-Escott problem. In some cases, this allows us to partition fewer than $b^N$ values into $b$ sets $\{S_1,\ldots,S_b\}$, such that $$\sum_{s\in S_1}s^k = \sum_{s\in S_2}s^k = \cdots = \sum_{s\in S_b}s^k$$ for $0\leq k \leq N-1$. The classical construction can only partition $b^N$ values such that the first $N$ powers agree. Our results are amenable to a computational search, which may discover new, smaller, solutions to this classical problem.