Text: We give an algorithm to enumerate all primitive abundant numbers (PAN) with a fixed Ω, the number of prime factors counted with their multiplicity. We explicitly find all PAN up to Ω=6, count all PAN and square-free PAN up to Ω=7 and count all odd PAN and odd square-free PAN up to Ω=8. We find primitive weird numbers (PWN) with up to 16 prime factors, the largest of which is a number with 14712 digits. We find hundreds of PWN with exactly one square odd prime factor: as far as we know, only five were known before. We find all PWN 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 PWN with a cubic (or higher power) odd prime factor, we prove that there are none with Ω≤7. We find several PWN with 2 square odd prime factors, and one with 3 square odd prime factors. These are the first such examples. We finally observe that these results are in favor of the existence of PWN with arbitrarily many prime factors. Video: For a video summary of this paper, please visit https://youtu.be/kEl20Tyf1zU. © 2019 Elsevier Inc.

Primitive abundant and weird numbers with many prime factors

Gianluca Amato
;
Maurizio Parton
2019

Abstract

Text: We give an algorithm to enumerate all primitive abundant numbers (PAN) with a fixed Ω, the number of prime factors counted with their multiplicity. We explicitly find all PAN up to Ω=6, count all PAN and square-free PAN up to Ω=7 and count all odd PAN and odd square-free PAN up to Ω=8. We find primitive weird numbers (PWN) with up to 16 prime factors, the largest of which is a number with 14712 digits. We find hundreds of PWN with exactly one square odd prime factor: as far as we know, only five were known before. We find all PWN 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 PWN with a cubic (or higher power) odd prime factor, we prove that there are none with Ω≤7. We find several PWN with 2 square odd prime factors, and one with 3 square odd prime factors. These are the first such examples. We finally observe that these results are in favor of the existence of PWN with arbitrarily many prime factors. Video: For a video summary of this paper, please visit https://youtu.be/kEl20Tyf1zU. © 2019 Elsevier Inc.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11564/691415
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