We solve multiple conjectures by Byszewski and Ulas about the base b sum-of-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 bN values into b sets $${\{S_1,\ldots,S_b\}}$$ { S 1 , … , S b } such that $$\sum_{s\in S_1}s^k = \sum_{s\in S_2}s^k = \cdots = \sum_{s\in S_b}s^k $$ ∑ s ∈ S 1 s k = ∑ s ∈ S 2 s k = ⋯ = ∑ s ∈ S b s k for $${0\leq k \leq N-1}$$ 0 ≤ k ≤ N - 1 . The classical construction can only partition bN 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.