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arXiv · 1001.4170

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

K. G. Hare·S. Laishram·T. Stoll
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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 $k$.

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K. G. Hare, S. Laishram, T. Stoll. 2010-01-23. The sum of digits of $n$ and $n^2$. https://arxiv.org/abs/1001.4170

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