Type: Article

On digit sums of multiples of an integer

Cécile Dartyge; Florian Luca; Pantelimon Stanica;

Abstract:

Let g > 1 be an integer and s(g)(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 s(g)(n) and s(g)(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 s(g)(n) = s(g)(kn) In the second part of the paper, we show that for any K > 0 the set of the integers n satisfying s(g)(n) <= K s(g)(kn) for all k is an element of N is of asymptotic density 0. This gives an affirmative answer to a question of W.M Schmidt (C) 2009 Elsevier Inc. All rights reserved