Theoretical Statistics and Mathematics Unit, ISI Delhi

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

by Kevin G. Hare, Shanta Laishram and Thomas Stoll

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 $k$.

isid/ms/2011/09 [fulltext]

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