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arXiv · 1802.07178

Primitive abundant and weird numbers with many prime factors

Gianluca Amato·Maximilian F. Hasler·Giuseppe Melfi·Maurizio Parton
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Abstract

We give an algorithm to enumerate all primitive abundant numbers (briefly, PANs) with a fixed $Ω$ (the number of prime factors counted with their multiplicity), and explicitly find all PANs up to $Ω=6$, count all PANs and square-free PANs up to $Ω=7$ and count all odd PANs and odd square-free PANs up to $Ω=8$. We find primitive weird numbers (briefly, PWNs) with up to 16 prime factors, improving the previous results of [Amato-Hasler-Melfi-Parton] where PWNs with up to 6 prime factors have been given. The largest PWN we find has 14712 digits: as far as we know, this is the largest example existing, the previous one being 5328 digits long [Melfi]. We find hundreds of PWNs with exactly one square odd prime factor: as far as we know, only five were known before. We find all PWNs with at least one odd prime factor with multiplicity greater than one and $Ω= 7$ and prove that there are none with $Ω< 7$. Regarding PWNs with a cubic (or higher) odd prime factor, we prove that there are none with $Ω\le 7$, and we did not find any with larger $Ω$. Finally, we find several PWNs with 2 square odd prime factors, and one with 3 square odd prime factors. These are the first such examples.

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Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton. 2018-02-27. Primitive abundant and weird numbers with many prime factors. https://doi.org/10.1016/j.jnt.2019.02.027

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