A strange attractor of primes

ONU McIntosh Center; Activities Room

The greatest prime factor sequences (GPF sequences), born at ONU in 2005, are integer sequences satisfying recursions in which every term is the greatest prime factor of a linear combination with positive integer coefficients of the preceding k terms (where k is the order of the sequence), possibly including a positive constant. The very first GPF sequence that was introduced satisfies the recursion x(n+1)=GPF(2*x(n)+1). In 2005 it was conjectured that no matter the seed, this particular GPF sequence enters the limit cycle (attractor) 3-7-5-11-23-47-19-13. In our current work, of a computational nature, we introduce the functions “depth” – where depth(n) is the number of iterations needed to reach the attractor given that the seed is x(0)=n, and “land” – where land(n) is the specific element of the attractor where the entrance is made, again, given that the seed is x(o)=n. We investigate the functions depth(n) and land(n), and calculate specific values using the MAPLE computer algebra system. This turns out to be a very interesting example of “arithmetic dynamics”.