On a Lehmer problem concerning Euler's totient function

Abstract

Let $M$ be a positive integer with $M > 4$, and let $\varphi$ denote Euler's totient function. If a positive integer $n$ satisfies the Diophantine equation (*) $M \varphi(n) = n - 1$, then the number of prime factors of $n$ is much bigger than $M$. Moreover, the set of all squarefree integers which do not fulfil (*) contains ``nice'' subsets.