3x+2

[y.kohmoto]

    On Mon, 9 Jun 2003 Edwin Clark wrote:

>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
>
>

    Dear Jud
    Thanks for the verifying up to 2,120,000.

    Dear Edwin
    I thought the same thing.
    "a prime p such that (f@@i)(p) is prime for i = 0, 1, 2, ...,"
    It looks like a Sophi German prime or Cuningham's prime chain.
    Is it true if your sequence is infinit then a counter example exists?



    Ralf Stephan wrote :
    >> %S A000001 3, 11, 7, 23, 71, 43, 131, 79, 239, 719, 127, 383, 1151,
691, 83,
    >> 251, 151,

    >until here they are all 3 mod 4, BTW.


    %S A000001 3, 11, 7, 23, 71, 43, 131, 79, 239, 719, 127, 383, 1151, 691,
83,
251, 151, 13, 41,
    %T A000001 5, 17, 53, 23, 71, 43, 131, 79, 239, 719, 127,

    Yes, and after here they are all 1 mod 4 until the first point of the
cycle.
    I don't know the reason.

    Yasutoshi