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

TL;DR

This paper generalizes the study of integers where the sum of digits in base q equals the sum of digits of their square or polynomial, extending previous results and analyzing specific cases.

Contribution

It extends Melfi's 2005 results to arbitrary bases and polynomials, providing a refined analysis and detailed case studies for quadratic polynomials.

Findings

Extended the characterization of numbers with equal digit sums for n and p(n) in general bases.

Provided a detailed analysis for the case p(n) = n^2, identifying subsets with fixed digit sum values.

Refined previous results by Melfi, offering new insights into the structure of such numbers.

Abstract

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 $s_2(n) = s_2(n^2)$. We extend this study to the more general case of generic $q$ 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 $k$.