Suppose we have a pseudorandom number generator PRNG with number of possible seed states K. Let us denote PRNG(k) the number yielded by the generator when the seed state is k. Here k is an integer between 1 and K.
Question A: Is it possible to adapt PRNG to yield pseudorandom integer numbers uniformly distributed between 1 and K?
Assuming that the answer to this question is yes, let's denote by PRNG(k; 1, K) the number between 1 and K yielded by the adapted PRNG when the seed state is k.
We can then build a new pseudorandom number generator by nesting PRNG, i.e.
newPRNG(k) = PRNG(PRNG(k; 1, K))
Question B: From a theoretical point of view, can we say something about the properties of the new PRNG? In particular, does this nesting trick increase or decrease the randomness of the generator, in some sense? I think this question is a natural one but I didn't find any answer on the web.