On digit sums of multiples of an integer
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Let g > 1 be an integer and Sᶢ (m) be the sum of digits in base g of the positive integer m. In this paper, we study the positive integers n such that sᶢ (n) and sᶢ (kn) satisfy certain relations for a fixed, or arbitrary positive integer k. In the first part of the paper, we prove that if n is not a power of g, then there exists a nontrivial multiple of n say kn such that sᶢ (n) = sᶢ (kn). In the second part of the paper, we show that for any K > 0 the set of the integers n satisfying sᶢ (n) ≤ Ksᶢ (kn) for all k ∈ ℕ is of asymptotic density 0. This gives an affirmative answer to a question of W.M. Schmidt.
The article of record as published may be found at http://dx.doi.org/10.1016/j.jnt.2009.04.003
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