On a problem of Nicol and Zhang
type: Item
instance of: scholarly article
title: On a problem of Nicol and Zhang
zbMATH Open document ID: 1241.11004
author: Florian Luca
author: József Sándor
published in: Journal of Number Theory
publication date: 2008-12-05T00:00:00Z
review text: The subject of the paper under review is studying the composite integers \(n\) which divide \(\varphi(n)+ \sigma(n)\). Let \(\mathcal{A}=\{n \text{ composite} : n\mid \varphi(n)+ \sigma(n)\}\), and recall that \(\Omega(n)\) denotes the total number of prime divisors of the integer \(n\), and \(\omega(n)\) denotes the number of its distinct prime divisors. The authors follow some arguments from elementary number theory to prove that there is no positive integer \(n\in \mathcal{A}\) with \(\omega(n)=2\), and also for any fixed positive integer \(K\geq 2\), the set \(\mathcal{A}\) contains only finitely many positive integers \(n\) with \(\Omega(n)\leq K\), the more as there are only finitely many odd positive integers \(n\in \mathcal{A}\) with \(\omega(n)=K\). In continuation, by using the ideas from the proof of the above results, the authors give a complete characterization of those \(n\in\mathcal{A}\) such that \(\omega(n)=3\). More precisely, they show that if \(n\in\mathcal{A}\) is such that \(\omega(n)=3\), then either \(n=2^\alpha.3.p\) with \(p=2^{\alpha-2}.7-1\) prime, or \(n\in\{560,588,1400\}\). However, in this paper, the authors cannot prove that there are infinitely many composite integers in \(\mathcal{A}\), but letting \(\mathcal{A}(x)=\mathcal{A}\cap [1,x]\) they prove that the estimate \[ \text{Card}(\mathcal{A}(x))\leq x\mathrm{e}^{-2^{-\frac12}(1+o(1))\sqrt{\log x\log\log x}},\qquad\text{as \(x\to\infty\)}, \] is valid. By applying partial summation formula this estimate implies that \(\sum_{n\in\mathcal{A}}1/n\) is convergent.
reviewed by: Mehdi Hassani
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 5378842
zbMATH Keywords: arithmetic function
zbMATH Keywords: Euler \(\varphi\) function
zbMATH Keywords: divisor \(\sigma\) function
zbMATH Keywords: asymptotic behavior
MaRDI profile type: Publication
full work available at URL: j.jnt.2007.03.012
OpenAlex ID: W2054382626
cites work: Some diophantine equations involving arithmetic functions
cites work: Handbook of Number Theory I
cites work: Q4272975
cites work: Q4854953
DOI: 10.1016/J.JNT.2007.03.012
description: scientific article; zbMATH DE number 5378842