On the conditional infiniteness of primitive weird numbers
type: Item
instance of: scholarly article
title: On the conditional infiniteness of primitive weird numbers
zbMATH Open document ID: 1364.11011
author: Giuseppe Melfi
published in: Journal of Number Theory
publication date: 2014-11-20T00:00:00Z
review text: 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 \[ \liminf_{n\to\infty}\frac{w_{n+1}-w_n}{w_n}=0. \]
reviewed by: Mehdi Hassani
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 6371490
zbMATH Keywords: divisor function
zbMATH Keywords: abundant number
zbMATH Keywords: semiperfect number
zbMATH Keywords: almost perfect number
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: j.jnt.2014.07.024
OpenAlex ID: W2093329006
Wikidata QID: Q55881540
cites work: The difference between consecutive primes. II
cites work: Q5301462
cites work: On Weird and Pseudoperfect Numbers
cites work: Sums of divisors and Egyptian fractions
cites work: Q3139663
cites work: Unsolved problems in number theory
cites work: Q4141292
cites work: Q3672063
cites work: Q4650315
cites work: Q4443440
DOI: 10.1016/J.JNT.2014.07.024
description: scientific article; zbMATH DE number 6371490