On the inverse problem relative to dynamics of the \(w\) function (Q1041571)

Let \(P\) be the set of prime numbers and \(P(n)\) denote the largest prime factor of integer \(n > 1\). Write \[ \begin{gathered} C_3 = \{ p_1 p_2 p_3 :p_i \in \mathcal{P}\,(i = 1,2,3),\,p_i \neq p_j\, (i \neq j)\}, \\ B_3 = \{ p_1 p_2 p_3 : p_i \in \mathcal{P}\,(i = 1,2,3),\,p_1 = p_2 \, \text{or} \, p_1 = p_3 \, \text{or} \, p_2 = p_3, \, \text{but not} \, p_1 = p_2 = p_3 \}.\end{gathered} \] For \(n = p _{1} p _{2} p _{3} \in C _{3} \cup B _{3}\), define the \(w\) function by \[ w(n) = P(p_1 + p_2 )P(p_1 + p_3 )P(p_2 + p_3 ). \] If there is \(m \in S \subset C _{3} \cup B _{3}\) such that \(w(m) = n\), then \(m\) is called \(S\)-parent of \(n\). It is proved that there are infinitely many elements of \(C_{3}\) which have enough \(C_{3}\)-parents and that there are infinitely many elements of \(B_{3}\) which have enough \(C_{3}\)-parents and also that there are infinitely many elements of \(B_{3}\) which have enough \(B_{3}\)-parents. ``Enough'' is explained in quantitative form. The proofs use a.o. a version of the large sieve and the prime number theorem.