Are there other pseudo-random distributions like the prime-numbers? Does there exists other structures in math, which are seemingly random, but deterministic, and follow rules similar to the prime numbers, by rules I mean there must be statements similar to goldbach's conjecture or twin-prime-conjecture etc, for instance i dont consider the digits of pi to have this sort of structure. Is there some field in math which deals with distributions which "seem" random, but still follow certain rules ?
What are some examples of similar structures?
Also, I want to know what are the necessary axioms to produce the prime-number distribution, and if one ca replace these axioms to get other pseudo-random distributions which are not "isomorphic" to the prime-numbers, and which are not constructable in regular axiomatic systems ?

The Ulam numbers are a good example of a seemingly "random" sequence, which certainly satisfies the Goldbach requirement (every positive integer is the sum of two Ulam numbers) and I believe it is conjectured that there are infinitely many pairs n, n+1 of Ulam numbers (I conjecture it, at least).
It is not hard to invent infinitely many sequences with sufficiently high natural density for which analogues to Goldbach and the Twin Prime conjectures can be expected to apply.  Give it a try!
You might enjoy Halberstam and Roth's book Sequences if you are interested in these sorts of features of sequences.
