On the congruence $\kappa n\equiv a(mod \varphi(n))$

Abstract

Lehmer's totient problem asks whether there exists a composite $n$ such that $\varphi(n)\mid (n-1)$, where $\varphi$ is the Euler's function. This problem is still open. Later, several upper bounds of the derived problem ``$\varphi(n)\mid (n-a)$'' were given. In this paper, we extend it to $n$ with $\varphi(n) \mid (\kappa n-a)$ and obtain some new bounds. As an application, for any integer $\lambda>0$ we have, \[ \#\{n\leq x: \varphi(n)\mid (n-1), n\not \equiv 1 (mod 6^{\lambda})\}\ll x^{1/2}/(\log{x})^{0.559552+\textit{o}(1)},\\ \#\{n\leq x: \varphi(n)\mid (3n-1)\} \ll x^{1/2}/(\log{x})^{0.559552+\textit{o}(1)}. \]