  In this paper, we prove that there are infinitely many $n$ for which
$rad(\varphi(n))|n-1$ but $n$ is not a Carmichael number. Additionally, we
prove that for any $k\geq 3$, there exist infinitely many $n$ such that
$\varphi(n)|(n-1)^k$ but $\varphi(n)\nmid (n-1)^{k-1}$. The constructs that we
consider here are generalizations of Carmichael and Lehmer numbers,
respectively, that were first formulated by Grau and Oller-Marc\'en.
