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Here is a sequence; experimental mathematics :

$2, 2, 2, 11, 11, 254908033,...$ we could define as :

Least primes such that

$((p+1)(nextprime(p)+1))-1$ is prime and

$((nextprime(p)+1)(nextprime(nextprime(p))+1))-1$ is prime and...

where chains of length $L= 1,2,3,4,5,...,n,...$ are formed.

As an example we do have $(2,3,11);(3,5,23);(5,7,47)$ ($L=3$ with $p=2$)

$[11, 13, 17, 19, 23, 29, 167, 251, 359, 479, 719]$

$[254908033, 254908037, 254908063, 254908097, 254908099, 254908109, 254908117, 64978106817377291, 64978114464618431, 64978129759102271, 64978138935793799, 64978141994690999, 64978146583036979]$

Again we can see the Guy law of small numbers. There is a huge gap between fifth and sixth term of the sequence. In other words very small primes get large $L$ values. Is this a finite sequence ? Are these kind of sequences useless ?

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That is such a programmer style. No mathematician would call a function "$\bf nextprime$". Ever. – Asaf Karagila Jan 31 '13 at 1:45
Yes, Asaf; though it is not a ambigous wording. I thought about it. In fact i am a very bad programmer, with not enough knowledge. To prove it here is the naive program i wrote in Pari gp: forprime(p=2,2*10^9,q=nextprime(p+1);r=nextprime(q+1);u=nextprime(r+1);x=nextpr‌​ime(u+1);z=nextprime(x+1);s=pq+p+q;t=rq+r+q;v=ru+r+u;y=xu+x+u;a=z*x+x+z;if(is‌​prime(s)&isprime(t)&isprime(v)&isprime(y)&isprime(a),print([p,q,r,u,x,z,s,t,v,y,a‌​]))) – user55514 Jan 31 '13 at 9:12
I wanted to write a shorter Pari gp program using vectors to keep the intermediate results of q and compare them with the previous p found; but i messed writing it. – user55514 Jan 31 '13 at 14:31

You ask three questions:

  1. Are sequences useful?

  2. Is this a finite sequence?

  3. Are these kind [sic] of sequences useless?

The answer to the first question is, yes.

The answer to the second question is, probably not, but probably beyond current knowledge, as are so many simply stated questions about primes. But perhaps I should first make sure I understand what these sequences are:

I think that given a positive integer $k$, you want $k$ consecutive primes, $p_1,p_2,\dots,p_k$, with $p_1$ as small as possible, such that $(p_i+1)(p_{i+1}+1)-1$ is prime for $i=1,2,\dots,k-1$. So, for example, for $k=6$, you have the consecutive primes $11,13,17,19,23,29$ with $$\displaylines{(11+1)(13+1)-1=167\cr(13+1)(17+1)-1=251\cr\dots\cr(23+1)(29+1)-1=719\cr}$$

Now there is something called Schinzel's Hypothesis H, which says, roughly speaking, that if you have a finite collection of multivariable polynomials, and there is no obvious reason why they shouldn't all simultaneously be prime, then they will all be simultaneously prime infinitely often. I think you could shoehorn this problem into Hypothesis H; it might be a good exercise to try it. Anyway, Hypothesis H is widely believed to be true, but is almost entirely unproved.

As to whether this kind of sequence is useful --- who knows? I doubt it, it's too special and too unmotivated and too far from anything known (to me) to be useful, but, who knows?

It does seem that others have already come across it. At it is reported that Jan van Delden & Farideh Firoozbakht found $8$ consecutive primes beginning with $5823417569$, and Giovanni Resta found $9$ starting at $3\,889\,108\,085\,627$.

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You may want to use \, or {,} so the number doesn't look like a sequence whose elements are $3$ and $889$ and $108$ and... – Rahul Jan 31 '13 at 1:16
@R, thanks, I had been wondering about a way to handle commas in big integers. – Gerry Myerson Jan 31 '13 at 1:30

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