There's a different sort of number theory, combinatorial, that benefits tremendously from computation and experimentation. An excellent website to get you started is http://www.experimentalmath.info/. Borwein and Bailey (responsible for that website) are two of the champions of using computers to find new results, and Zeilberger is the godfather of using computers to prove combinatorial results.

Surprisingly, of the various branches of number theory (analytic, algebraic, combinatorial, additive), it seems to me that analytic number theory has benefited the least from experimentation.