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

Abstract

Büyükaşik et al. [Publ. Math. Debrecen, 2024] introduced a family of generalizations of Euler's totient function $\ddot{I}†(n)$, by setting $\ddot{I}{†}_k(n)={\sum }_a{a}^k$ for $a\in [1,n]$ such that $(a,n)=1$, with $\ddot{I}{†}_0(n)=\ddot{I}†(n)$. Letting ${\mathcal{D}}_s=\{k\ge s:\forall n\ge 1\ddot{I}{†}_s(n)\mid \ddot{I}{†}_k(n)\}$, Büyükaşik et al. proved that ${\mathcal{D}}_s$ is finite for each $s\ge 0$, 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.

Submitted for publication