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    $\begingroup$ That's a very important observation! I suspect this will be quite useful for later answers. $\endgroup$
    – Peter
    Commented Aug 22, 2023 at 5:22
  • $\begingroup$ A possible observation: in certain sequences where (at least it appears) the factorisation p_0*p_1*p_3*p’ works for any prime p > some amount (for example, 2*2*5*p’ for p’ > 10), possibly excluding the case p’ = p_n. In those cases it appears the “some amount” is the product p_0p_n. For example, I would conjecture that other than 3*5*11*11, the next such number that fits this category in the sequence is 3*5*11*37. This would indicate that the rule may correlate its largest prime with the others (though this may just be a coincidence, as there are exceptions). $\endgroup$
    – yanjunk
    Commented Aug 23, 2023 at 3:59
  • $\begingroup$ Note also that while the 4-2 and 5-2 sequences both start at p' = 17 (presumably after 2^4), the 6-2 and 7-2 sequences start at 37 (presumably after 2^5). $\endgroup$
    – yanjunk
    Commented Aug 23, 2023 at 4:11
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    $\begingroup$ A rule very close to making sense is that one of the two middle factors (which multiply to the number) is a power of a prime. The rule might involve the two middle factors in some way. $\endgroup$
    – yanjunk
    Commented Aug 23, 2023 at 4:44