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 $e$-th power. We say that $x$ is a base-$b$, $e$-power, height-$h$, $u$-attracted number if $h$ is the smallest positive integer such that $S_{e,b}^h\left(x\right)=u$. Happy numbers are then base-10, 2-power, 1-attracted numbers of any height. Let ${\sigma }_{h,e,b}\left(u\right)$ denote the smallest height-$h$, $u$-attracted number for a fixed base $b$ and exponent $e$ and let $g\left(e\right)$ denote the smallest number such that every integer can be written as $x_1^e+x_2^e+\cdots +x_{g\left(e\right)}^e$ for some nonnegative integers $x_1,x_2,\dots ,x_{g\left(e\right)}$. We prove that if $p_{e,b}$ is the smallest nonnegative integer such that $b^{p_{e,b}}>g\left(e\right)$,

$$d=\lceil \frac{g\left(e\right)+1}{1-{\left(\frac{b-2}{b-1}\right)}^e}+e+p_{e,b}\rceil ,$$

and ${\sigma }_{h,e,b}\left(u\right)\ge b^d$, then $S_{e,b}\left({\sigma }_{h+1,e,b}\left(u\right)\right)={\sigma }_{h,e,b}\left(u\right)$.