On a Lehmer problem concerning Euler's totient function (Q1434808)

\textit{D. H. Lehmer} [Bull. Am. Math. Soc. 38, 745--751 (1932; Zbl 0005.34302)] asked whether there exists any composite number \(n\) such that \(\varphi (n)| n-1\), that is, (*) \(M \varphi (n)=n-1\) for some \(M\). This question is still open. The present authors review some facts concerning (*) presented in the literature and show that if \(n\) satisfies (*) with \(M>4\), then the number of prime factors of \(n\) is much greater than \(M\), and that the set of all squarefree integers which do not fulfil (*) contains ``nice'' subsets.