Composite numbers n whose sum of prime factors is prime

Abstract

In this dissertation, we will look for positive integers n such that β(n) = P p|n p = P, where P is prime. We write p, P to indicate prime num bers. The function β(n) has already been studied by a few authors. Its first appearance was in the papers of Pomerance and his co–authors, including Erd˝os (see [9], [13], [14]), where the positive integers n with β(n) = β(n+ 1) were investigated. An example of such n is 714, a number which at that time appeared in the context of a baseball game, and since then such num bers have been called Ruth–Aaron numbers. De Koninck and Luca studied positive integers n with β(n) | n (see [5], [6]). This thesis is based on a paper by Luca and Moodley, which estimated the number of composite integers n ≤ x such that β(n) is a prime. An example of this is 210 where β(210) = 17. We shall now proceed in starting this thesis formally. The main result of the paper which this thesis is based on is as follows: Let B = {n composite : β(n) is prime}. For a subset A of positive integers and a real number x ≥ 1, we write A(x) = A ∩ [1, x]. We have the following theorem. Theorem 1.1. The estimates x (log x) 3 #B(x) x log x hold [12]. 1 We believe the upper bound is closer to the truth. We can prove a lower bound of the same order of magnitude as the upper bound assuming a uniform version of the Bateman–Horn Conjecture. We will first provide some definitions needed to read the proof of the main result, motivation for the study of this question, and methods used in answering this question.