Saidak's proof on infintely many primes defines this recursive series: $n_{i+1}=n_{i}*(n_{i}+1)$. If $n_1=2$, we get 2, 6, 42, 1806, 3263442 etc. The number of prime factors in this series is 1, 2, 3, 4, 6 etc. Is there an estimate on this latter series?
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$\begingroup$ The products miss the prime 5 and likely miss other primes of the form 6k-1. So I suspect the latter series does not double after the second term, and probably grows as loglog of the pronic term, which growth is slightly faster than the growth of the index. Gerhard "Not Seeing Much Smoothness Here" Paseman, 2018.04.07. $\endgroup$– Gerhard PasemanCommented Apr 7, 2018 at 18:59
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3$\begingroup$ This is A091336 in the OEIS. $\endgroup$– LSpiceCommented Apr 7, 2018 at 19:00
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$\begingroup$ And 2, 6, 42, 1806, 3263442, ... is A007018, which also contains a link to an MO question $\endgroup$– მამუკა ჯიბლაძეCommented Apr 7, 2018 at 23:13
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