A tree approach to the happy function

TL;DR

This paper introduces a tree-based method to construct and analyze $e$-power $b$-happy numbers of any height, providing formulas for their enumeration and exploring their relationships within a structured framework.

Contribution

It presents a novel tree construction technique to encode $e$-power $b$-happy numbers, their heights, and ancestry relations, along with formulas for their preimages.

Findings

Constructed a tree encoding happy numbers and their relations.

Derived a formula for the cardinality of preimages of the happy function.

Applied the technique to analyze unhappy numbers of a given height.

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.