Elated Numbers

TL;DR

This paper investigates the properties of the $b$-elated function, including fixed points, cycles, and sequences of $b$-elated numbers, highlighting differences from the related $b$-happy function.

Contribution

It characterizes fixed points and cycles of the $b$-elated function and explores sequences and minimal heights, revealing behaviors distinct from the $b$-happy function.

Findings

Determined fixed points and cycles of $E_{2,b}$.

Analyzed sequences of $b$-elated numbers and their minimal heights.

Showed notable differences between $b$-elated and $b$-happy functions.

Abstract

For a base $b \geq 2$, the $b$-elated function, $E_{2,b}$, maps a positive integer written in base $b$ to the product of its leading digit and the sum of the squares of its digits. A $b$-elated number is a positive integer that maps to $1$ under iteration of $E_{2,b}$. The height of a $b$-elated number is the number of iterations required to map it to $1$. We determine the fixed points and cycles of $E_{2,b}$ and prove a range of results concerning sequences of $b$-elated numbers and $b$-elated numbers of minimal heights. Although the $b$-elated function is closely related to the $b$-happy function, the behaviors of the two are notably different, as demonstrated by the results in this work.