On $k$-Lehmer numbers

Abstract

Lehmer's totient problem consists of determining the set of positive integers $n$ such that $\phi (n)|n-1$ where $\phi$ is Euler's totient function. In this paper we introduce the concept of $k$-Lehmer number. A $k$-Lehmer number is a composite number such that $\phi (n)|(n-1{)}^k$. The relation between $k$-Lehmer numbers and Carmichael numbers leads to a new characterization of Carmichael numbers and to some conjectures related to the distribution of Carmichael numbers which are also $k$-Lehmer numbers.