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 $S_{ e,b}(\sigma_{ h+1,e,b}(u))=\sigma_{ h,e,b}(u)$.
Categories: math.NT