Note: I'm currently in a low level algebra class and have very little knowledge of some of the more complex mathematical concepts. That being said, I can probably figure out anything I don't know through research.
If I understand correctly, prime numbers become (roughly)exponentially less common and cannot be calculated readily with an explicit formula. Mainly what I'm interested is the fact that they become increasingly difficult to calculate and are more difficult to calculate than to verify(verifying 7 is prime is quicker than finding all prime numbers between 1 and 10). Is there another set of numbers like this, or are primes absolutely unique in this sense? I don't really care what the ratio is, as long as finding these numbers takes longer than it takes to verify them and cannot be found with an elementary function.