On the conditional infiniteness of primitive weird numbers
Let \(n\) be a positive integer. Let \(\sigma(n)\) denote the sum of divisors of \(n\). If \(\sigma(n)>2n\), then \(n\) is called abundant. If \(n\) can be expressed as a sum of distinct proper divisors of \(n\), then \(n\) is called semiperfect. A weird number is an abundant number that is not semiperfect, and is called primitive if it is not a multiple of other weird numbers. In the paper under review, the author proves that if \(p=2^{k+2}-a\) and \(q=2^{k+2}+b\) are primes, for some positive integer \(k\) and positive odd integers \(a\) and \(b\) satisfying \(b+3<a<2^{(k-1)/2}\), then \(n=2^kpq\) is a primitive weird number. The author ends the paper by proposing two following conjectures: Conjecture 1. There are infinitely many primitive weird numbers of the form \(2^kpq\), with \(p\) and \(q\) primes. Conjecture 2. Let \((w_n)_{n\geq 1}\) be the sequence of primitive weird numbers. Then \[ \liminf_{n\to\infty}\frac{w_{n+1}-w_n}{w_n}=0. \]
scientific article; zbMATH DE number 6371490