We study prime divisors of various sequences of positive integers A ( n ) + 1 , n = 1 , … , N , such that the ratios a ( n ) = A ( n ) / A ( n − 1 ) have some number-theoretic or combinatorial meaning. In the case a ( n ) = n , we obviously have A ( n ) = n ! , for which several new results about prime divisors of n ! + 1 have recently been obtained.