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arXiv · 2606.01633

On a problem on a generalization of Euler's totient function

John M. Campbell
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Abstract

B\"uy\"uka\c{s}ik et al. [Publ. Math. Debrecen, 2024] introduced a family of generalizations of Euler's totient function $\varphi(n)$, by setting $\varphi_k(n) = \sum_{a} a^k$ for $a \in [1, n]$ such that $(a, n) = 1$, with $\varphi_0(n) = \varphi(n)$. Letting $\mathcal{D}_{s} = \{ k \geq s : \forall n \geq 1 \ \varphi_s(n) \mid \varphi_k(n) \}$, B\"uy\"uka\c{s}ik et al. proved that $\mathcal{D}_{s}$ is finite for each $s \geq 0$, and conjectured that $\mathcal{D}_{1} = \{ 1, 3, 15 \}$ and provided computations to support this conjecture. We succeed in proving this conjecture, using an argument based on our extensive interactions with GPT-5.5 Pro.

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John M. Campbell. 2026-06-01. On a problem on a generalization of Euler's totient function. https://arxiv.org/abs/2606.01633

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