The Korselt set of a power of a prime

Abstract

A Carmichael number is a composite number [Formula: see text] such that [Formula: see text] divides [Formula: see text] for all integers [Formula: see text] coprime to [Formula: see text]. Korselt discovered that [Formula: see text] is a Carmichael number if and only if [Formula: see text] is square-free and [Formula: see text] for each prime divisor [Formula: see text] of [Formula: see text]. Let [Formula: see text], a [Formula: see text]-number is defined to be a composite number [Formula: see text], such that [Formula: see text] and [Formula: see text] for each prime [Formula: see text]. The set of all [Formula: see text] such that [Formula: see text] is a [Formula: see text]-number is called the Korselt set of [Formula: see text] and we denote this set by [Formula: see text]. In this paper, we investigate some properties of [Formula: see text] when [Formula: see text] is a power of a prime.