3x+2

[Edwin Clark]

On Sun, 8 Jun 2003, Jud McCranie wrote:

> At 05:41 AM 6/8/2003, y.kohmoto wrote:
> %N A000001 a(n) is the greatest prime factor of 3*a(n-1)+2 .
> 
> >     I conjectured as follows :
> >     "for all number m, if a(1)=m then the sequence becomes cyclic."
> >
> >     Is there any counter example?
> 
> A quick test shows it holds to 2,120,000 (at least).
> 
> 

I got similar results. Note that one just needs to check it for
primes. This raises the question: Let f(x)=3x+2. Is it true that for each
n there is a prime p such that (f@@i)(p) is prime for i = 0, 1, 2, ...,
n. [Here f@@i is the composition of f with itself i times.]

 If a(n) = the least such prime then we have the sequence 

    2, 3, 5, 29, 1129, 10009, 575119

For example, this means that if p = 29 then p,f(p),f(f(p)),f(f(f(p))) are
primes.  

Is there another term in this sequence? 

Has anyone seen another such sequence with a different f?

--Edwin