On Arithmetic Progressions of Integers with a Distinct Sum of Digits

Let b ≥ 2 be a fixed integer. Let sb(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 sb(n), sb(n+ q), . . . , sb(n+ q(k − 1)) are (pairwise) distinct. More specifically, let Lb,q denote the supremum of k as n varies in the set of nonnegative integers N. We show that Lb,q is bounded from above and hence finite. Then it makes sense to define μb,q as the smallest n ∈ N such that one can take k = Lb,q. We provide upper and lower bounds for μb,q. Furthermore, we derive explicit formulas for Lb,1 and μb,1. Lastly, we give a constructive proof that Lb,q is unbounded with respect to q.