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arXiv · 2501.10616

The Forestry of Adversarial Totient Iterations

Luis Palacios Vela·Christian Wolird
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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.

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Luis Palacios Vela, Christian Wolird. 2025-01-18. The Forestry of Adversarial Totient Iterations. https://arxiv.org/abs/2501.10616

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