[seqfan] Collatz-like problem with prime iterations

Vladimir Shevelev shevelev at bgu.ac.il
Sat Oct 29 15:50:06 CEST 2011

 Let P(n) be the maximal prime divisor of 3*n+1. For a given n, consider iterations p_1=P(n), p_2=P(p_1),

..., p_(k+1)=P(p_(k)),... I believe that, for every n, some iterate equals 2, when we stop the process.
E.g., for n=27, we have the following primes: 41,31,47,71,107,23,7,11,17,13,5,2. To every n>=1corresponds a set of primes.
The sequence of  products of primes over these sets begins with  2, 170170, 10, 130, 2, 13394810, 24310,10,170170, etc.
Let t=t(n) be the smallest number of iterates such that P^t(n)<n (n>=3). The first numbers t(n) are t(3)=2, t(4)=3, t(5)=1, t(6)=6, t(7)=4, t(8)=1, t(9)=1, t(10)=6,...(I submitted A198724).
       Is this sequence bounded? Even if the answer is in negative, the sequence of places  of records of this sequence, most likely,  grows very fast.


 Shevelev Vladimir‎

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