coudy's answer, pointing you to the Experimental Mathematics, is the right answer, and there are already other MO questions about experimental mathematics that are relevant, but I can't resist giving some examples.

• Here's one example that has a similar flavor to yours, but with the Riemann hypothesis instead of the Collatz conjecture. In his Math. Comp. paper, On the distribution of spacings between zeros of the zeta function, Andrew Odlyzko reported on a computational study of Montgomery's pair correlation conjecture. There are of course many other examples in Math. Comp.; often, the criterion for publication here is some kind of algorithmic novelty (e.g., new algorithmic techniques had to be devised in order to push the computation out much further than before).

• A workhorse of experimental mathematics is the empirical discovery of relationships between real numbers by applying lattice basis reduction algorithms to their decimal expansions. Some of these may be proved, while others remain conjectural. A sample paper that I like is About a New Kind of Ramanujan-Type Series, by Jesús Guillera, which among other things presents Gourevitch's conjecture: $$\sum_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7 = \frac{32}{\pi^3}.$$

• Siemion Fajtlowicz's paper On Conjectures of Graffiti reported on a program that searched empirically for graph-theoretic conjectures. Of course, most were either trivial or false or known, but some turned out to be new conjectures which were not trivial to prove.