Suppose we where given a super computer that would be capable of computing anything, but only for one day. We could for instance compute many of the Ramsey numbers. What would be some computations that would expand our knowledge of mathematics, and would give us results that would be used by many other mathematicians?
Example of an answer:
The Ramsey number $R(n,m)$ is the smallest integer $j$ such that $K_j$ must always contain a monochromatic red $K_n$ or a monochromatic blue $K_m$ when its edges are colored red or blue. A lot of effort has been made to calculate the Ramsey numbers. But even "small" Ramsey numbers like $R(5,5)$ are not well known (although it is known to be between $43$ and $49$). However these bounds become a lot worse for even slightly smaller Ramsey numbers like $R(10,9)$ for which the bound is $581-12677$. The computer could then be used to calculate a few thousand Ramsey numbers.