An interesting property of this sequence: if5, 3, 5, 3, 3, 11, 13, 13, 37, 17, 11, 17, 7, 5, 5, 17, 7, 17, 3, 17, 5, 31 (1)

*is the greatest prime factor function, then its infinite periodic extension*

**P****(X(n))**satisfies the recursion

The intriguing part: for virtually all random choices of the initial conditions that I made - that is, other than the trivial ones (leading to 1-cycles, such as those withX(n)=P[X(n-1)+3X(n-2)+2X(n-3)] (2)

*X(0)=X(1)=X(2)=p*), the prime sequences satisfying

**(2)**ultimately enter the same limit cycle

**(1)**. These are strange phenomena, of a similar nature to the

**GPF-tribonacci sequences**(where we discovered that there are at least

*four*distinct GPF-tribonacci limit cycles, of lengths 100, 212, 28 and 6). My last

*MAPLE*experiment with

**(2)**involved a seed consisting of three 80 digits primes,

**X(0) = 67525204474446805798439049565857966823399463795779492274451732592404979647691781**

**X(1) = 43280708928363322606959208124229795876921456250899031972019611375562107511756173**

**X(2) = 50558523494317177773504742464927157574684415344640904005922668959550436919047547**

which eventually led to the cyclic shift of

**(1)**given by

**7, 17, 3, 17, 5, 31, 5, 3, 5, 3, 3, 11, 13, 13, 37, 17, 11, 17, 7, 5, 5, 17**, with

**7=X(50), 17=X(51), 3=X(52), ...**

*Added on April 17, 2012*

**(1)**is the only one found. Today - improved the process of random selection of initial conditions. A sample in the last series of searches - still leading to

**(1)**(log plot of the graph included below)

X(0)=38790912184195861716665094754005872283807697220069306337918513393335402427949730921338400505241

X(1)=34380504590591201002245563

X(2)=239197136656882070249024044251770704653065407229190712881313724790604984709544507179663