On a problem of Nicol and Zhang
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.
scientific article; zbMATH DE number 5378842