On Arithmetic Progressions of Integers with a Distinct Sum of Digits

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,L-q denote the supremum of k as n varies in the set of nonnegative integers N. We show that L-b,L-q is bounded from above and hence finite. Then it makes sense to define mu(b,q) as the smallest n is an element of N such that one can take k = L-b,L-q. We provide upper and lower bounds for mu(b,q). Furthermore, we derive explicit formulas for L-b,L-1 and mu(b,1). Lastly, we give a constructive proof that L-b,L-q is unbounded with respect to q.