Fixed Points of Augmented Generalized Happy Functions II: Oases and Mirages

TL;DR

This paper investigates the existence and properties of fixed points in augmented generalized happy functions, focusing on sets of parameters called oases where fixed points occur, and provides bounds and examples for small bases.

Contribution

It introduces the concept of oases base b, analyzes their properties, and computes bounds and minimal examples for small bases.

Findings

Defined k-oasis base b as parameter sets with fixed points

Established basic properties of oases base b

Computed bounds and minimal examples for small bases

Abstract

An augmented generalized happy function $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base $b$ digits plus $c$. For $b\geq 2$ and $k \in \mathbb{Z}^+$, a $k$-desert base $b$ is a set of $k$ consecutive non-negative integers $c$ for each of which $S_{[c,b]}$ has no fixed points. In this paper, we examine a complementary notion, a $k$-oasis base $b$, which we define to be a set of $k$ consecutive non-negative integers $c$ for each of which $S_{[c,b]}$ has a fixed point. In particular, after proving some basic properties of oases base $b$, we compute bounds on the lengths of oases base $b$ and compute the minimal examples of maximal length oases base $b$ for small values of $b$.