The Forestry of Adversarial Totient Iterations

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