seq lookup 5 29 1129 10009 a(n)=smallest of n primes by x= 3*x+2.

[Don McDonald]

On Sun, 8 Jun 2003  Edwin Clark   wrote -
: Subject: Re: 3x+2 :
...

:  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

I recently posted sequence A081173   (extended to 14 terms...)
a(n) = max prime factor of  a(n-1)^2+2, which may grow erratically.

ID
Number:  A081173 Sequence:
2,3,11,41,17,97,3137,13499,60741001,14158633,7424699571433,  
  18375387908679124623224497,152868746152697352174823427,  
  114585848725150699093848122619332057  
Name:  a(1) = 2, then a(n) = greatest prime factor of a(n-1)^2+2.  
Example:  a(2) = 3 because 3 is greatest prime factor of
2^2+2. a(3)=11 because  

************
however, here is my PARI program for Edwin C sequence.

? mx=1;forprime(p=1,198000,x=p;m=0;while(isprime(x=3*x+2),m=m+1;
if(m>mx,print(p,"  ",m, "  ",x);mx=m,)))

Table
start/ number of primes in succession?/ end prime of run.
       by x = 3x+2.   [error =1.]

5  2  53
29  3  809
1129  4  91529
10009  5  2432429        should be 6 primes in succn**** x=p;m=1;

sequence lookup 5 29 1129 10009   ***************
a(n) =smallest of  n primes in succession by
x = 3*x+2.

a(3) = 5 because  5 is prime,
5*3+2= 17 is prime and 17*3+2 = 53 is prime.
5 is the smallest of 3 primes in succession, x= 3x +2.
*******************
compare Cunningham chain. p= 2*prime+1.
2 5 11 23 47..

Sorry, Edwin Clark usf, seqfan.
Edwin has already obtained the sequence [not returned by superseeker]
and (1969 reference) to articles on Lehmer or Shanks.

If not already, Edwin may be partly credited with this sequence.

Note. (3x oddprime+2) is never a multiple of 3, or 2.
 If (3x+2) is  composite, its least prime factor may be
 at least 5 and its greatest prime
factor should be less than x.

Consider the least prime p, such that the sequence
x = max prime factor of (3x+2)
does not cycle.



Do the (3x+2) 's have to be all prime thereafter??

NO.  But they have to be greater than p,
else p would start with a smaller  prime!

An exercise is to find the smallest x within the respective
cycles of    x = max prime factor if (3x+2.)

I hope to find ...a Pari program for ..
 while(max prime factor(x=3x+2) is not less than (p).), please.

don.mcdonald.
>  .user.PariGp.3prim+2    11.06.03  23:55


working.
warning. Please delete or skip.

==========
> 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.]
: 


? mx=1;forprime(p=1,198000,x=p;m=0;while(isprime(x=3*x+2),m=m+1;
if(m>=mx,print(p,"  ",m, "  ",x);mx=m,)))

Table.
start/ number of primes in run xx error=1./ last prime in run.
         [ equal or better record.]

3  1  11
5  1  17
5  2  53
7  2  71
19  2  179
29  2  269
29  3  809
139  3  3779
1129  3  30509
1129  4  91529
10009  4  810809
10009  5  2432429




*gp             GP/PARI CALCULATOR Version 1.36
                 (Archimedes version)


stacksize = 1000000, prime limit = 200000, buffersize = 30000

? v=[]; forprime(p=1,200, if(isprime(3*p+2),v=concat(v,3*p+2),));v

Cunningham chain, prime = 3xprime+2.

%1 = [11, 17, 23, 41, 53, 59, 71, 89, 113, 131, 179, 239, 251, 269, 293, 311, 383, 419, 449, 491, 503, 521, 593, 599]


? v=[]; forprime(p=19*10^4,19*10^4+500,
if(isprime(3*p+2),v=concat(v,3*p+2),));v 
%4 = [570083, 570191, 570389, 570851, 571019, 571163, 571211, 571229]

?
?  forprime(p=19*10^4,19*10^4+500,print( factor(3*p+2)) )

factors [max prime factor] of 3x prime +2.

[570083, 1]
[5, 1; 17, 1; 19, 1; 353, 1]
[5, 1; 114031, 1]
[570191, 1]
[37, 1; 15413, 1]
[83, 1; 6871, 1]
[5, 1; 114073, 1]
[570389, 1]
[661, 1; 863, 1]
...

? forprime(p=10000,10080,x=p;while(isprime(x=3*x+2),print(p, "  ", x)))
10009  30029
10009  90089
10009  270269
10009  810809
10009  2432429       ******** record up to 190 000.
10037  30113
10039  30119
10039  90359
10039  271079
10067  30203

? 6 primes in progression xxx  3x+2.

*sp.