#### Vol. 11, No. 2, 2018

 Download this article For screen For printing  Recent Issues  The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
Numbers and the heights of their happiness

### May Mei and Andrew Read-McFarland

Vol. 11 (2018), No. 2, 235–241
##### 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 ${b}^{{p}_{e,b}}>g\left(e\right)$,

$d=⌈\frac{g\left(e\right)+1}{1-{\left(\frac{b-2}{b-1}\right)}^{e}}+e+{p}_{e,b}\phantom{\rule{0.3em}{0ex}}⌉,$

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

##### Keywords
happy numbers, integer sequences, iteration, integer functions
##### Mathematical Subject Classification 2010
Primary: 11A99, 11A63
##### Milestones 