Vol. 11, No. 2, 2018

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Numbers and the heights of their happiness

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