Let $p\_l$ the $l$-th prime number. I've considered the formula 


$$\frac{N\_{n+1}}{N\_n}+\frac{N\_{n+2}}{p\_{n+1}N\_n}\pm1$$
where $N\_k=\prod\_{l=1}^k p\_l$ is the primorial of order $k$. Previous formula can be simplified thus as $$p\_{n+1}+p\_{n+2}\pm 1.\tag{1}$$


I wondered, for $n\geq 1$ running over positive integers, when the terms of previous sequence provide us twin prime pairs $$(p\_{n+1}+p\_{n+2}-1, p\_{n+1}+p\_{n+2}+1).\tag{2}$$



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> **Question.** Let $p\_n$ be the $n$-th prime number. Is it possible to prove that there exist finitely many twin prime pairs of the form $$(p\_{n+1}+p\_{n+2}-1, p\_{n+1}+p\_{n+2}+1)$$ when $n$ runs over positive integers ? **Many thanks.**
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The last paragraph of the section *Properties* from the Wikipedia [*Twin prime*](https://en.wikipedia.org/wiki/Twin_prime#Properties) tell us the elementary sieve that satisfy the sequence of twin primes. I don't know if it, or Brun's theorem, can be useul.


**Computational evidence.** 1) My belief is that the formula $(1)$ provide us many twin prime pairs, but I don't know if it is a remarkable proportion of those. You can evaluate the code (it is a line) using the web Sage Cell Server (choose *GP* as language)


`for(n=1, 100, if(isprime(prime(n+1)+prime(n+2)-1)==1&&isprime(prime(n+1)+prime(n+2)+1)==1,print(prime(n+1)+prime(n+2)-1)))`


And you can do a comparison using this other code


`for(n=1, 881, if(isprime(n)==1&&isprime(n+2)==1,print(n)))`


Previous outputs are thus terms of the sequence *A001359*, from the OEIS. You can to evaluate the code for different segments of integers $1\leq n\leq N$, and choose a different output from the function `print()`


2) My belief is that this sequence/s aren't in the literature.



On the quantity of twin prime pairs of a given form
Indeed, standard conjectures imply that there should be infinitely many such twin prime pairs even if we insist that $p\_{n+1},p\_{n+2}$ are themselves twin primes! In other words, there should be infinitely many integers $p$ such that $p$, $p+2$, $p+(p+2)-1=2p+1$, and $p+(p+2)+1=2p+3$ are all prime—this is a special case of Schinzel's Hypothesis H (and almost a special case of the Hardy–Littlewood conjectures prime $k$-tuples conjecture).


