Numbers and the Heights of their Happiness

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 $S_{e,b}^h(x)=u$. 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 $x_1^e+x_2^e+\mathrm{.}\mathrm{.}\mathrm{.}+x_{g(e)}^e$ for some nonnegative integers $x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_{g(e)}$. 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 ${\sigma }_{h,e,b}(u)\ge b^d$, then $S_{e,b}({\sigma }_{h+1,e,b}(u))={\sigma }_{h,e,b}(u)$.