Distribution of primes and dynamics of the ω function

Let P be the set of all primes. The following result is proved: For any nonzero integer a, the set a + P contains arbitrarily long sequences which have the same largest prime factor. We give an application to the dynamics of the in function which extends the "seven" in Theorem 2.14 of [Wushi Goldring, Dynamics of the omega function and primes, J. Number Theory 119 (2006) 86-98] to any positive integer. Beyond this we also establish a relation between a result of congruent covering systems and a question on the dynamics of the omega function. This implies that the answer to Conjecture 2.16 of Goldring's paper is negative. Two conjectures are posed. (C) 2008 Elsevier Inc. All rights reserved.