OEIS solves real-world problems! One time I needed to optimize accesses to a data structure in my code. By trying out a few examples, I found the optimal order of iterations followed a particular number sequence, but I had no idea on how it could be deduced or calculated analytically. I entered the few examples I had into OEIS and found the solution. The solution involves graduate-level combinatorics and there was no way that I could solve it.
Thanks Neil Sloane and OEIS.