Let be the greatest prime factor of the integer x. We associate a class of sequences of primes to any pair of positive integers. The sequences are defined recursively in the following way: (p is a prime) and for We prove that all sequences in are ultimately periodic and provide computational evidence suggesting the conjecture that for any pairof positive integers and any sequence in is ultimately periodic. For example, all the sequences ultimately enter the period 5, 11, 23, 47, 19, 13, 3, 7 for all choices of

greatest prime factor, recurrences, periodic, primes, sequences.