# [seqfan] Re: Collatz-like problem with prime iterations

franktaw at netscape.net franktaw at netscape.net
Sun Oct 30 06:03:44 CET 2011

```The sequence is almost certainly unbounded. Just as there are believed
to be arbitrarily long sequences of primes with p_(n+1) = 2*p_n + 1
(the same for 2*p_n - 1), there are almost certainly arbitrarily long
sequences of primes with p_(n+1) = (3*p_n + 1) / 2.

-----Original Message-----
From: Vladimir Shevelev <shevelev at bgu.ac.il>

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.

Regards,