In the Big-O analysis being able to stop the outer loop early turns out to be a red herring. Being able to stop the inner loop after 1/p operations is key, as is the density of the primes. It means that the work is O(n*(sum of 1/p)). The sum of 1/i scales like log(n), the density of the primes scales as 1/log(n), and between them they cancel out for O(n*log(log(n))).

log(log(n)) is essentially constant in the range of numbers I can test before hitting memory limitations. Also there is a theoretical log(n) that we are missing from the overhead of addressing and representing ever larger numbers. While we tend to call that constant, in reality it is not.

BTW if you are interested, longer tables of maximal gaps have been compiled...