I suppose asymptotics for certain functions (e.g., Prime Number Theorem), or any sort of conjecture based on large empirical evidence, would count, but that's probably not what you mean.
Perhaps more interesting is the following. In high school/college, I was briefly interested in automated theorem proving and read about this (I don't remember the source and may miff the details--perhaps someone can help). Around the 60's or 70's, someone wrote an AI program to use numerical evidence to have a computer "deduce" many theorems/conjectures in number theory. They showed their answers to Knuth, and he marked the ones he thought were mathematically interesting. At least one thing that stood out was several interesting "results" on highly composite numbers, which I think Ramanujan may have studied as well.