Primitive abundant and weird numbers with many prime factors

We give an algorithm to enumerate all primitive abundant numbers (briefly, PANs) with a fixed \(\Omega\) (the number of prime factors counted with their multiplicity), and explicitly find all PANs up to \(\Omega=6\), count all PANs and square-free PANs up to \(\Omega=7\) and count all odd PANs and odd square-free PANs up to \(\Omega=8\). 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 \(\Omega = 7\) and prove that there are none with \(\Omega < 7\). Regarding PWNs with a cubic (or higher) odd prime factor, we prove that there are none with \(\Omega\le 7\), and we did not find any with larger \(\Omega\). 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.