On the classification of integers \(n\) that divide \(\varphi(n)+\sigma(n)\)
type: Item
instance of: scholarly article
title: On the classification of integers \(n\) that divide \(\varphi(n)+\sigma(n)\)
zbMATH Open document ID: 1185.11008
author: Kelley Harris
published in: Journal of Number Theory
publication date: 2009-07-06T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 11N56
zbMATH DE Number: 5575279
zbMATH Keywords: special arithmetical functions
zbMATH Keywords: divisibility of these
reviewed by: Jürgen Spilker
MaRDI profile type: Publication
full work available at URL: j.jnt.2009.02.003
OpenAlex ID: W2093403350
cites work: Q5317673
cites work: More on the total number of prime factors of an odd perfect number
cites work: On a problem of Nicol and Zhang
cites work: Some diophantine equations involving arithmetic functions
DOI: 10.1016/J.JNT.2009.02.003
description: scientific article; zbMATH DE number 5575279