On the densities of some subsets of integers (Q930754)

In this note, the author presents nice proofs of two conjectures concerning densities of subsets of positive integers suggested in [[1] \textit{A. Feist}, Missouri J. Math. Sci. 15, No. 3, 173--177 (2003; Zbl 1162.11302)] and [[2] \textit{N. E. Elliott} and \textit{D. Richner}, Missouri J. Math. Sci. 15, No. 3, 189--199 (2003; Zbl 1161.11324)], respectively. 1) Following [1], a positive integer \(n\) is called a sigma-prime if \(n\) and \(\sigma(n)\) are coprimes, where \(\sigma(n)\) is the sum of the divisors of \(n\). Let \(\mathcal{SP}\) be the set of all sigma-primes. It was conjectured by Feist that \(\mathcal{SP}\) is of asymptotic density zero. Here, this conjecture is proved. Theorem 1. The inequality \(\#\mathcal{SP}(x)\ll \frac{x}{\log \log \log x}\) holds for all \(x>e^e\). Some remarks concerning phi-primes (substituting \(\sigma(n)\) by Euler's function \(\varphi(n)\)) are made. 2) Following [2], a positive integer \(n\) is called an ans number if it admits a representation of the form \(p^2-q^2\), where \(p\) and \(q\) are primes. Let \(\mathcal{ANS}\) denote the set of all ans numbers. It was conjectured by Elliott and Richner that \(\mathcal{ANS}\) is of asymptotic density zero. Here, this conjecture is proved. Theorem 2. The inequality \(\#\mathcal{ANS}(x) \ll \frac{x}{\log x}\) holds for all \(x>1\).