On twin EP numbers

Abstract

EP numbers were introduced by Estrada and Pogliani in 2008. These are positive integers $E(n)$ defined as the product of $n$ and the sum of the digits of $n$. Estrada and Pogliani suspected that there may be infinitely many twin EP numbers; i.e. those pairs in this sequence that differ by one. It is relatively easy to show that three consecutive EP numbers do not exist, and that no pair $E(n)$ and $E(m)$ can be twins for infinitely many bases $b$. The main contribution of our work is the result that indeed there are infinitely many twin EP numbers over any base. The proof is constructive and makes use of elementary properties of natural numbers. The forms of the twin EP numbers presented are derived from continued fractions. The behavior of the series of the reciprocals of twin EP numbers is also considered.