Numbers and the heights of their happiness

Abstract

A generalized happy function, $S_{e,b}$ maps a positive integer to the sum of its base $b$ digits raised to the $e^\text{th}$ power. We say that $x$ is a base $b$, $e$ power, height $h$, $u$ attracted number if $h$ is the smallest positive integer so that $S^{h}_{e,b}(x)=u$. Happy numbers are then base 10, 2 power, 1 attracted numbers of any height. Let $\sigma_{h,e,b}(u)$ denote the smallest height $h$, $u$ attracted number for a fixed base $b$ and exponent $e$ and let $g(e)$ denote the smallest number so that every integer can be written as $x_{1}^{e}+x_{2}^{e}+...+x_{g(e)}^{e}$ for some nonnegative integers $x_{1},x_{2},...,x_{g(e)}$. In this paper we prove that if $p_{e,b}$ is the smallest nonnegative integer such that $b^{p_{e,b}}>g(e)$, $\displaystyle d=\left\lceil \frac{g(e)+1}{1-(\frac{b-2}{b-1})^{e}}+e+p_{e,b}\right\rceil$, and $\sigma_{h,e,b}(u)\geq b^{d}$, then $S_{e,b}(\sigma_{h+1,e,b}(u))=\sigma_{h,e,b}(u)$.