Fang 2022 searching boundary abundance odd weird numbers

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theorem_1_1: Fang's exhaustive computer search, run on the volunteer computing project yoyo@home in 2013-2015, finds no odd weird number below 10^21.

theorem_1_2: Fang's restricted computer search finds no odd weird number N below 10^28 whose abundance sigma(N) - 2N is smaller than 10^14.

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Wenjie Fang, Searching on the boundary of abundance for odd weird numbers. arXiv:2207.12906 (2022). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2207.12906), every other right reserved.

A number is weird if it is abundant but not pseudoperfect (no subset of its proper divisors sums to it). Theorem 1.1 (p. 2) establishes by exhaustive computation that no odd number below 10^21 is weird, extending Hearn's earlier 10^17 bound recorded in OEIS A006037. Theorem 1.2 (p. 2) pushes the range much further under a restriction on abundance: no odd number below 10^28 with abundance sigma(N) - 2N smaller than 10^14 is weird. The method is a depth-first search on a tree of numbers built prime by prime from their factorizations. It skips the descendants of pseudoperfect numbers, since the smallest odd weird number is primitive abundant (Proposition 3.1), and of numbers whose descendants below the bound are all deficient (Proposition 3.3); the author views it as a simple application of reverse search from combinatorial optimization. The computation was done with the volunteer computing project yoyo@home during 2013-2015. The paper bears on problem 470, the Benkoski-Erdős question of whether an odd weird number exists, by providing a computational lower bound, 10^21, on any such number rather than a resolution.

Source: https://arxiv.org/abs/2207.12906. The copy read for this card is arXiv:2207.12906v1 (26 July 2022); the page numbers above are that preprint's.

Read status: claims checked for the results linked below, statements read clause by clause on the printed pages of arXiv v1; the computation is not rerun and no proof is checked step by step.

Bears on.

• #470: the problem's first question asks whether an odd weird number exists. Theorem 1.1 shows none lies below 10^21, and Theorem 1.2 excludes those below 10^28 with abundance smaller than 10^14; both are computational lower bounds and do not answer the question. The problem's second question, whether there are infinitely many primitive weird numbers, is recalled on pp. 1--2 and not addressed.

Results.

• Theorem 1.1 (p. 2): "There is no odd weird number below 10^21."
• Theorem 1.2 (p. 2): "There is no odd weird number below 10^28 with abundance smaller than 10^14." Here the abundance of N is A(N) = sigma(N) - 2N (p. 1).
• Method (Sections 2--3, pp. 3--6): A depth-first search on the tree in which the parent of N is N divided by its largest prime factor, pruned by Propositions 3.1 and 3.3; the author calls it an application of reverse search that may be useful for other searches for integers with factorization-dependent properties.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.