Numbers and the Heights of their Happiness

TL;DR

This paper studies generalized happy functions based on digit sums raised to a power, exploring properties of attracted numbers, their heights, and their relation to number representations across different bases and exponents.

Contribution

It introduces new theoretical results linking the heights of attracted numbers to their digit sum properties and number representations in various bases and exponents.

Findings

Established bounds on the height of attracted numbers.

Proved relationships between digit sum functions and number representations.

Derived conditions for the stability of attracted numbers under the happy function.

Abstract

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