The sum of digits of $n$ and $n^2$

Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 2005, Melfi examined the structure of $n$ such that $s_2(n) = s_2(n^2)$\mathrm{.}Weextendthisstudytothemoregeneralcaseofgeneric$q$ and polynomials $p(n)$, and obtain, in particular, a refinement of Melfi’s result. We also give a more detailed analysis of the special case $p(n)=n^2$, looking at the subsets of $n$ where $s_q(n)=s_q(n^2)=k$ for fixed $k$.