Papers › Primitive abundant and weird numbers with many prime factors
Primitive abundant and weird numbers with many prime factors
Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, Maurizio Parton
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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≤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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