Dynamics of Goldringʼs w-function

TextLet A3A3 be the set of all positive integers pqr, where p, q, r   are primes such that at least two of them are not equal. Denote by P(n)P(n) the largest prime factor of n  . For n=pqr∈A3n=pqr∈A3, define w(n):=P(p+q)P(p+r)P(q+r)w(n):=P(p+q)P(p+r)P(q+r). In 2006, Wushi Goldring proved that for any n∈A3n∈A3, there exists an i   such that wi(n)∈{20,98,63,75}wi(n)∈{20,98,63,75}, where w0(n)=nw0(n)=n and wi(n)=w(wi−1(n))wi(n)=w(wi−1(n)) (i⩾1i⩾1). If w(m)=nw(m)=n, then m is called a parent of n  . Let B3B3 be the set of all positive integers pq2pq2 of A3A3. In this paper, we study the function w   extensively. For example, one of our results is that there exist infinitely many n∈B3n∈B3 which have at least n1.1886n1.1886 parents in B3B3. Several open problems are posed.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=8FWvR8_KoHA.