The Forestry of Adversarial Totient Iterations

TL;DR

This paper derives a closed-form expression for nested Euler's totient functions and explores their boundedness for specific integer sequences like squares and cubes, introducing the Arboreal Algorithm for potential closed-form solutions.

Contribution

It provides a new closed-form expression for complex nested totient functions and introduces the Arboreal Algorithm to analyze their structure.

Findings

Nested totient functions are bounded for sequences of squares and cubes.

The Arboreal Algorithm can sometimes find closed forms of these functions.

The paper extends understanding of iterated totient functions and their properties.

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.