  Lehmer's totient problem consists of determining the set of positive integers
$n$ such that $\varphi(n)|n-1$ where $\varphi$ 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 $\varphi(n)|(n-1)^k$. 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.
