Let b ≥ 2 be a fixed integer. Let s_b(n) denote the sum of digits of the nonnegative integer n in the base-b representation. Further let q be a positive integer. In this paper we study the length k of arithmetic progressions n, n + q, ..., n + q(k-1) such that s_b(n), s_b(n + q), ..., s_b(n + q(k-1)) are (pairwise) distinct. More specifically, let L_b,q denote the supremum of k as n varies in the set of nonnegative integers N. We show that L_b,q is bounded from above and hence finite. Then it makes sense to define μ_b,q as the smallest n ∈ N such that one can take k = L_b,q. We provide upper and lower bounds for μ_b,q. Furthermore, we derive explicit formulas for L_b,1 and μ_b,1. Lastly, we give a constructive proof that L_b,q is unbounded with respect to q.