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On Arithmetic Progressions of Integers with a Distinct Sum of Digits

2012
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Journal of Integer Sequences
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unpublished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.

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