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

TL;DR

This paper demonstrates the existence of infinitely many numbers with specific divisibility properties related to Euler's totient function, extending the concepts of Carmichael and Lehmer numbers in number theory.

Contribution

It proves the infinitude of generalized Lehmer numbers not being Carmichael and extends the understanding of divisibility properties of Euler's totient function for higher powers.

Findings

Infinitely many $n$ with $rad({ ext{phi}}(n))|n-1$ but not Carmichael.

Existence of infinitely many $n$ with ${ ext{phi}}(n)|(n-1)^k$ but ${ ext{phi}}(n) mid (n-1)^{k-1}$ for $k extgreater=3$.

Generalization of Carmichael and Lehmer numbers concepts.

Abstract

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.