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arXiv · 1908.02194

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

Breeanne Baker Swart·Susan Crook·Helen G. Grundman·Laura Hall-Seelig·May Mei·Laurie Zack
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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 $b$.

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Breeanne Baker Swart, Susan Crook, Helen G. Grundman, Laura Hall-Seelig, May Mei, Laurie Zack. 2019-08-06. Fixed Points of Augmented Generalized Happy Functions II: Oases and Mirages. https://arxiv.org/abs/1908.02194

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