On primitive weird numbers of the form $2^k p q$

TL;DR

This paper classifies all primitive weird numbers of the form 2^k p q with p<q primes for 1≤k≤14, expanding the known set of such numbers and providing a comprehensive analysis of their structure.

Contribution

It systematically finds all primitive weird numbers of the form 2^k p q for specified k and introduces new larger primitive weird numbers of this form.

Findings

Complete classification for 1≤k≤14

Discovery of larger primitive weird numbers

Expanded the known set of primitive weird numbers

Abstract

We say a natural number~$n$ is abundant if $\sigma(n)>2n$, where $\sigma(n)$ denotes the sum of the divisors of~$n$. The aliquot parts of~$n$ are those divisors less than~$n$, and we say that an abundant number~$n$ is pseudoperfect if there is some subset of the aliquot parts of~$n$ which sum to~$n$. We say~$n$ is weird if~$n$ is abundant but not pseudoperfect. We call a weird number~$n$ primitive if none of its aliquot parts are weird. We find all primitive weird numbers of the form $2^kpq$ ($p<q$ being odd primes) for $1\le k\le14$. We also find primitive weird numbers of the same form, larger than any previously published.