KORSELT NUMBERS AND SETS

Abstract

Let α ∈ ℤ\{0}. A positive integer N is said to be an α-Korselt number (Kα-number, for short) if N ≠ α and p - α divides N - α for each prime divisor p of N. We are concerned, here, with both a numerical and theoretical study of composite squarefree Korselt numbers. The paper contains two main results. The first one shows that for α ∈ ℤ\{0}, the following properties hold: (i) If α ≤ 1, then each composite squarefree Kα-number has at least three prime factors. (ii) Suppose that α > 1. Let p < q be two prime numbers and N ≔ pq. If N is an α-Korselt number, then p < q ≤ 4α - 3. In particular, there are only finitely many α-Korselt numbers with exactly two prime factors. Let α ∈ ℕ\{0}; by an α-Williams number (Wα-number, for short) we mean a positive integer which is both a Kα-number and a K-α-number. Our second main result shows that if p, 3p - 2, 3p + 2 are all prime, then their product is a (3p)-Williams number.