On digit sums of multiples of an integer

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 ( k n ) 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 ( k n ) . 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 ( k n ) for all k ∈ N is of asymptotic density 0. This gives an affirmative answer to a question of W.M. Schmidt.