What about the (diagonal) Ramsey Numbers, $k \to R(k,k)$? These are fairly natural to define but notoriously hard to compute. Indeed, only the first two Ramsey numbers $R(3,3)=6$ and $R(4,4)=18$ are known. We do know that $R(5,5) \in [43,49]$, which is the subject of the following joke (I may be butchering the content).
Joke. If an omnipotent alien came down to Earth and demanded that we determine $R(5,5)$ within a week, then mankind should divert all of its brainpower and resources to achieve this goal.
If on the other hand, the omnipotent alien demanded that we determine $R(6,6)$ within a week, then mankind should diverta divert all of its brainpower and resources to destroy the alien.