Ibrahim Al-Rasasi, King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics PO Box 5046, Dhahran 31261, Saudi Arabia e-mail: irasasi@kfupm.edu.sa

Othman Echi, King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics PO Box 5046, Dhahran 31261, Saudi Arabia e-mail: echi@kfupm.edu.sa; othechi@yahoo.com

Nejib Ghanmi, Umm Al-Qura University, University College in Makkah, Department of Mathematics, Azizia PO.Box 2064, Makkah, Kingdom of Saudi Arabia e-mail: naghanmi@uqu.edu.sa; neghanmi@yahoo.fr

Let \(\alpha\in \mathbb{Z}\setminus \{0\}\). A positive composite squarefree integer \(N\) is said to be an \(\alpha\)-Korselt number (\(K_{\alpha}\)-number, for short) if \(N\neq \alpha\) and \(p-\alpha\) divides \(N-\alpha\) for each prime divisor \(p\) of \(N\). By the Korselt set of \(N\), we mean the set of all \(\alpha\in \mathbb{Z}\setminus \{0\}\) such that \(N\) is a \(K_{\alpha}\)-number. This set will be denoted by \(\mathcal{KS}(N)\).

In a recent paper , Bouallegue–Echi–Pinch have asked whether there are infinitely many squarefree composite numbers with empty Korselt set. This paper aims to solve this question by showing that for each prime number \(q\geq 19\), \(6q\) has an empty Korselt set.

We also show that for each integer \(l\geq 3\), there are infinitely many squarefree composite numbers with \(l\) prime divisors whose Korselt sets are empty.

Soit \(N\) un nombre composé sans facteur carré et \(\alpha\in \mathbb{Z} \setminus \{0\}\). On dit que \(N\) est \(\alpha\)-Korselt si \(N\neq \alpha\) et \(p-\alpha\) divise \(N-\alpha\) pour tout facteur premier \(p\) de \(N\).

L’ensemble constitué de tous les \(\alpha\) tels que \(N\) est \(\alpha\)-Korselt, noté \(\mathcal{KS}(N)\), est appelé l’ensemble de Korselt de \(N\).

Bouallegue–Echi–Pinch se sont posés la question d’existence d’une infinité de nombres composés sans facteur carré possédant des ensembles de Korselt vides.

Dans ce papier on donne une réponse positive à cette question en démontrant que pour tout premier \(q\geq 19\), \(6q\) a un ensemble de Korselt vide.

On prouve aussi que pour tout entier \(l\geq 3\), il existe une infinité de nombres composés sans facteur carré ayant \(l\) facteurs premiers et d’ensembles de Korselt vides.