Numbers and the heights of their happiness

Mei, May; Read-McFarland, Andrew

doi:10.2140/involve.2018.11.235

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Numbers and the heights of their happiness

May Mei and Andrew Read-McFarland

Vol. 11 (2018), No. 2, 235–241

DOI: 10.2140/involve.2018.11.235

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

d = ⌈ \frac{g (e) + 1}{1 - {(\frac{b - 2}{b - 1})}^{e}} + e + p_{e, b} ⌉,

and $σ_{h, e, b} (u) \geq b^{d}$ , then $S_{e, b} (σ_{h + 1, e, b} (u)) = σ_{h, e, b} (u)$ .

Keywords

happy numbers, integer sequences, iteration, integer functions

Mathematical Subject Classification 2010

Primary: 11A99, 11A63

Milestones

Received: 4 November 2015

Revised: 5 April 2017

Accepted: 9 May 2017

Published: 17 September 2017

Communicated by Kenneth S. Berenhaut

Authors

May Mei
Department of Mathematics & Computer Science Denison University Granville, OH United States

Andrew Read-McFarland
Department of Mathematics & Computer Science Denison University Granville, OH United States

Vol. 11, No. 2, 2018