Bounding the number of solutions of some congruences

We estimate from above the number of solutions in integers n of congruence equations A(n) = λ (mod p), y < n ≤ x for various sequences A(1), . . . , A(N). Here p is a prime, x, y are integers, and each of our sequences under consideration is such that a(n) = A(n)/A(n -1) has certain prescribed arithmetic or algebraic properties. In particular, we deal with the situations that a(n) is a polynomial function of re, the n-th prime, or that a_n has some combinatorial meaning.