Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.1

On Arithmetic Progressions of Integers with a Distinct Sum of Digits

Carlo Sanna


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 nN 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.

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(Concerned with sequence A000120.)

Received August 5 2012; revised version received September 23 2012. Published in Journal of Integer Sequences, October 2 2012.

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