The sum of digits of n and n^2

La somme des chiffres de n et n^2

Kevin G. Hare (1) , Shanta Laishram (2) , Thomas Stoll (3)

1 University of Waterloo [Waterloo]
2 Statistics and Mathematics Unit
3 Université de la Méditerranée - Aix-Marseille 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). 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.

Domaines: Théorie des nombres [math.NT]

Kevin G. Hare, Shanta Laishram, Thomas Stoll. The sum of digits of n and n^2. International Journal of Number Theory, 2011, 7, pp.1737-1759. ⟨hal-01278745⟩