It’s not persistent if it can’t be recovered later

"Store data persistently" implies "it can be looked up" since if you cannot look it up it is impossible to know if it is stored persistently.

The "efficiently" part can be considered a separate problem though.

Well, if you just want to store data, you can use files. Lookup is a bit tedious and inefficient.

So, if we consider that persistent storage is a solved problem, then we can say that the reason for databases was how to look up data efficiently. In fact, that is why they were invented, even if persistent storage is a prerequisite.

How about "store data in certain way." That sounds more like 1 problem and encompasses an even larger problem space.