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arXiv · 2410.13990

A tree approach to the happy function

Eva G. Goedhart·Yusuf Gurtas·Pamela E. Harris
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Abstract

In this article, we present a method to construct $e$-power $b$-happy numbers of any height. Using this method, we construct a tree that encodes these happy numbers, their heights, and their ancestry--relation to other happy numbers. For fixed power $e$ and base $b$, we consider happy numbers with at most $k$ digits and we give a formula for the cardinality of the preimage of a single iteration of the happy function. We show that these happy numbers arise naturally as children of a given vertex in the tree. We conclude by applying this technique to $e$-power $b$-unhappy numbers of a given height.

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Eva G. Goedhart, Yusuf Gurtas, Pamela E. Harris. 2024-10-17. A tree approach to the happy function. https://arxiv.org/abs/2410.13990

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