Dynamics of the w function and primes

We begin by defining a function w on the setA3={n=p1e1⋯pses∈Z>1|∑i=1sei=3,ei>0,s>1}, where pipi is prime and pi≠pjpi≠pj for i≠ji≠j. If n∈A3n∈A3 then was can write n=pqrn=pqr where p, q, r are primes and possibly two, but not all three of them are equal. For any positive integer m  , let P(m)P(m) be its largest prime factor. Define the function w   on A3A3 byw(n)=w(pqr)=P(p+q)P(p+r)P(q+r).w(n)=w(pqr)=P(p+q)P(p+r)P(q+r). Our goal is to study the dynamics of w  . One of our main results is that every element of A3A3 is periodic with period a cyclic permutation of the period of 20.