A tree approach to the happy function

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.

Figures (8)

• Figure 1: The first two levels of the tree $\mathcal{T}_{2,30}$.
• Figure 2: The three levels of the finite tree $\mathcal{T}_{e,2}^5$.
• Figure 3: The three levels of the finite tree $\mathcal{T}_{2,3}^4$.
• Figure 4: The three levels of $\mathcal{T}_{3,3}^4$.
• Figure 5: First level of the finite graphs $\mathcal{C}^3_{2, 7}[10]$ and $\mathcal{C}^3_{2, 7}[25]$.

Theorems & Definitions (18)

• Remark
• Example 2.1
• Example 2.2
• Remark
• Example 3.1
• Proposition 3.1.1
• proof
• Lemma 3.2
• proof
• Proposition 3.2.1