On the classification of integers \(n\) that divide \(\varphi(n)+\sigma(n)\)
An integer \(n>1\) satisfies \(\varphi(n)+\sigma(n)=2n\) iff \(n\) is prime. Let \[ {\mathcal A}:=\left\{n\in\mathbb N:\frac{\varphi(n)+\sigma(n)}{n} \in\mathbb N_{\geq 3} \right\},\quad{\mathcal A}_k:=\{n\in{\mathcal A}:\omega(n)= k\}. \] \textit{F. Luca} and \textit{J. Sándor} proved [J. Number Theory 128, No. 4, 1044--1059 (2008; Zbl 1241.11004)], that every \(n\in{\mathcal A}\) has at least 3 prime-factors. The author presents a computer-implementable algorithm to decide, whether \({\mathcal A}_k\) contains only even integer (``Nicol's conjecture'' [\textit{C. A. Nicol}, J. Math. Anal. Appl. 15, 154--161 (1966; Zbl 0139.27202)]). He proves the conjecture for \(k=5\) and 6, finds the structure of \({\mathcal A}_3\) and \({\mathcal A}_4\) and shows that \(\{n \in{\mathcal A}:\Omega(n)\) fixed\} and \(\{n\in{\mathcal A}_k:n\,\text{odd}\) are finite sets.
scientific article; zbMATH DE number 5575279