Størmer's Theorem provides a method to find all consecutive p-smooth numbers. In Lehmer's paper On a problem of Størmer, a very weak upper bound is given (Theorem 7). M. F. Hasler observes at A002072 that, for $n$ primes, $10^n/n$ is an upper bound, except for $n=4$. This is in my view a very strong bound. Is there a proof of this result or a stronger bound than Lehmer's somewhere?
I made this graph to demonstrate how tight this bound is (extraordinarily close for a few small $n$):
