Primitive weird numbers having more than three distinct prime factors

TL;DR

This paper investigates the structure of primitive weird numbers with multiple prime factors, providing conditions and algorithms for their generation, including potential methods for finding odd weird numbers, an open problem.

Contribution

It introduces new sufficient conditions and algorithms for generating primitive weird numbers with more than three prime factors, extending previous methods and addressing the open question of odd weird numbers.

Findings

Algorithms for generating primitive weird numbers with multiple prime factors

Conditions ensuring a number is weird based on its factorization

Potential approaches for discovering odd weird numbers

Abstract

In this paper we study some structure properties of primitive weird numbers in terms of their factorization. We give sufficient conditions to ensure that a positive integer is weird. Two algorithms for generating weird numbers having a given number of distinct prime factors are presented. These algorithms yield primitive weird numbers of the form $mp_1\dots p_k$ for a suitable deficient positive integer $m$ and primes $p_1,\dots,p_k$ and generalize a recent technique developed for generating primitive weird numbers of the form $2^np_1p_2$. The same techniques can be used to search for odd weird numbers, whose existence is still an open question.