How to control false positives in sequential A/B testing while keeping a low sample size?
=run a maximum of n tests or stop earlier if a good improvement is found


Don't do that. Don't stop before n tests have been run if you like what you've seen so far....

http://www.evanmiller.org/how-not-to-run-an-ab-test.html

In general, however, figuring out how big an N you need given an expected effect size and alpha level is the domain of power analysis. Read more here:

http://www.uta.edu/faculty/sawasthi/Statistics/stpowan.html