Let p be the set of prime numbers and P(n) denote the largest prime factor of integer n > 1. Write C3={p1p2p3: pi ∈(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 w(n) = P(p1 + p2)P(p1 + p3)P(p2 + p3). If there is m ∈ S C3 ∪ B3 such that w(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.
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