Searching on the boundary of abundance for odd weird numbers

TL;DR

This paper reports extensive computational searches for odd weird numbers, providing evidence of their non-existence up to very large bounds, and introduces a novel reverse search method applicable to similar number theory problems.

Contribution

The authors conducted large-scale searches for odd weird numbers up to 10^{28} and developed a reverse search approach to optimize the search process.

Findings

No odd weird numbers found up to 10^{21}

Extended search up to 10^{28} with no odd weird numbers

Introduced a reverse search method for combinatorial optimization in number theory

Abstract

Weird numbers are abundant numbers that are not pseudoperfect. Since their introduction, the existence of odd weird numbers has been an open problem. In this work, we describe our computational effort to search for odd weird numbers, which shows their non-existence up to $10^{21}$. We also searched up to $10^{28}$ for numbers with an abundance below $10^{14}$, to no avail. Our approach to speed up the search can be viewed as an application of reverse search in the domain of combinatorial optimization, and may be useful for other similar quest for natural numbers with special properties that depend crucially on their factorization.