Sums of Digits in q-ary Expansions

Abstract

Let $s_q(n)$ denote the sum of the digits of a number $n$ expressed in base $q$. We study here the ratio $\frac{s_q(n^{\alpha })}{s_q(n)}$ for various values of $q$ and $\alpha$. In 1978, Kenneth B. Stolarsky showed that $\lim {\inf }_{n\to \infty }\frac{s_2(n^2)}{s_2(n)}=0$ and that $\lim {\sup }_{n\to \infty }\frac{s_2(n^2)}{s_2(n)}=\infty$ using an explicit construction. We show that for $\alpha =2$ and $q\ge 2$, the above ratio can in fact be any positive rational number. We also study what happens when $\alpha$ is a rational number that is not an integer, terminating the resulting expression by using the floor function.

18 pages

References in corpus (1)

• On simultaneous binary expansions of $n$ and $n^2$