Infinitude of $k$-Lehmer numbers which are not Carmichael

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