具有多个prime factors的本原富足奇异数
Primitive abundant and weird numbers with many prime factors
摘要：
我们给出了一个算法来枚举所有的原始丰富数（简称PAN）与一个固定的$Ω$（素因子的数量与他们的重数计数），并显式地找到所有的PAN到$Ω=6$，计数所有的PAN和无平方PAN到$Ω=7$和计数所有的奇PAN和奇无平方PAN到$Ω=8$。我们发现原始奇怪的号码（简称PWNs）多达16个素因子，改善以前的结果[Amato-Hasler-Melfi-Parton]其中PWNs多达6个素因子已经给出。我们发现的最大的PWN有14712位数：据我们所知，这是现有的最大的例子，前一个是5328位长[Melfi]。我们发现了数百个PWN，其中恰好有一个平方奇素因子：据我们所知，以前只知道五个。我们发现所有的PWNs至少有一个奇素因子的重数大于1和$Ω= 7$，并证明没有与$Ω< 7$。关于PWNs与立方（或更高）奇素数因子，我们证明，有没有与$Ω\le 7$，我们没有发现任何较大的$Ω$。最后，我们发现了几个PWNs的2平方奇素因子，和一个3平方奇素因子。这是第一个这样的例子。
英文摘要：
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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