Sums of digits in q-ary expansions

Abstract

Let sq(n) denote the sum of the digits of a number n expressed in base q. We study here the ratio [Formula: see text] for various values of q and α. In 1978, Kenneth B. Stolarsky showed that [Formula: see text] and that [Formula: see text] using an explicit construction. We show that for α = 2 and q ≥ 2, the above ratio can in fact be any positive rational number. We also study what happens when α is a rational number that is not an integer, terminating the resulting expression by using the floor function.