On the inverse problem relative to dynamics of the ω function

Let P be the set of prime numbers and P (n) denote the largest prime factor of integer n > 1. Write C3 = {p1p2p3 : pi ∈ P (i = 1, 2, 3), pi = pj (i = j)}, B3 = {p1p2p3 : pi ∈ P (i = 1, 2, 3), p1 = p2 or p1 = p3 or p2 = p3, but not p1 = p2 = p3}. For n = p1p2p3 ∈ C3 ∪ B3, we define the w function by ω(n) = P (p1 + p2)P (p1 + p3)P (p2 + p3). If there is m ∈ S - C3 ∪ B3 such that ω(m) = n, then we call m S-parent of n. We shall prove that there are infinitely many elements of C3 which have enough C3-parents and that there are infinitely many elements of B3 which have enough C3-parents. We shall also prove that there are infinitely many elements of B3 which have enough B3-parents.