The Forestry of Adversarial Totient Iterations

TL;DR

The paper studies the scoreboard function $A^\varphi(n)=\varphi(a_1+\varphi(a_2+\cdots+\varphi(a_n)))$ under adversarial iterations between Euler's totient function and a chosen sequence $A$. It develops inductive upper bounds and the Arboreal Algorithm to analyze totient fibers and totient trees, yielding closed-form descriptions for the scoreboard when $A$ is the positive integers or the perfect squares. For the cubes, it provides numerical evidence that no closed form exists and that multiple trees survive, underscoring the variability with growth rate. The work highlights a framework for bounding and understanding iterated totients, with potential generalizations to polynomial-growth sequences and connections to the probabilistic model $\mathbb{E}[\varphi(n)]=\frac{6}{\pi^2}n$. The results shed light on how the growth rate and parity density of $A$ influence the boundedness and structure of $A^\varphi(n)$.

Abstract

We give a closed-form expression for $\varphi(1+\varphi(2+\varphi(3+...+\varphi(n)$, where $\varphi$ is Euler's totient function. More generally, for an integer sequence $A=\{a_j\}$ we study the value of $A^\varphi(n)=\varphi(a_1+\varphi(a_2+\varphi(a_3+...+\varphi(a_n)$ when $A$ is the perfect squares or the perfect cubes. We show $A^\varphi(n)$ is bounded for all sequences considered. We also present the Arboreal Algorithm which can sometimes determine a closed form of $A^\varphi(n)$ using tree-like structures.

Figures (10)

• Figure 1: The adversarial iteration for $\mathbb{N}^\varphi(10)$.
• Figure 2: The totient tree produced by the Arboreal Algorithm for $\mathbb{N}^\varphi(n)=4$.
• Figure 3: Progression of the upper bound on partial evaluations.
• Figure 4: The totient trees corresponding to all possible values that the partial evaluations might take for $\mathbb{N}^\varphi(n)=1$, $\mathbb{N}^\varphi(n)=2$, and $\mathbb{N}^\varphi(n)=6$. Red nodes mark the numbers that are not in the image of $\varphi$. Orange nodes mark numbers that are ruled out by the previous upper bounds. And green nodes mark the legal values for $n$ we're looking for (which the authors like to imagine as fruit hanging from the trees).
• Figure 5: A portion of the totient tree corresponding to $A^\varphi(n)=4$ where $A=(1,4,9,...)$ is the perfect squares.

Theorems & Definitions (16)

• Lemma 2.1
• proof
• Corollary 2.1.1
• proof
• Corollary 2.1.2
• proof
• Theorem 2.2
• proof
• Lemma 4.1
• proof