Searching on the boundary of abundance for odd weird numbers

07/26/2022

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by   Wenjie Fang, et al.

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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.