On $k$-Lehmer numbers

TL;DR

This paper introduces $k$-Lehmer numbers, a generalization of Lehmer's totient problem, and explores their connection to Carmichael numbers, providing new characterizations and conjectures about their distribution.

Contribution

The paper defines $k$-Lehmer numbers and establishes their relationship with Carmichael numbers, offering new insights and conjectures in number theory.

Findings

New characterization of Carmichael numbers involving $k$-Lehmer numbers

Conjectures on the distribution of Carmichael numbers that are also $k$-Lehmer numbers

Introduction of the concept of $k$-Lehmer numbers as a generalization of Lehmer's problem

Abstract

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.