John Charles Saunders

Sums of Digits in q-ary Expansions

Let $s_q(n)$ denote the sum of the digits of a number $n$ expressed in base $q$. We study here the ratio $\frac{s_q(n^α)}{s_q(n)}$ for various values of $q$ and $α$. In 1978, Kenneth B. Stolarsky showed that $\lim\inf_{n\rightarrow\infty}\frac{s_2(n^2)}{s_2(n)}=0$ and that $\lim\sup_{n\rightarrow\infty}\frac{s_2(n^2)}{s_2(n)}=\infty$ using an explicit construction. We show that for $α=2$ and $q\geq 2$, 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.