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