On Arithmetic Progressions of Integers with a Distinct Sum of Digits

Carlo Sanna
2012 Journal of Integer Sequences   unpublished
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.
more » ... e 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.