The conjectural density of twin primes is $\frac {c\cdot n}{(\log n)^2}$ at a $c>0$.


Consider integers of form $p,p+1=2^tq,p+2=r$ where $p,q,r$ are primes and $t\geq1$ holds. 


1. Is there any reason to believe there are infinite of them at a given $t\geq1$? Is there a conjectural density for such triples at a given $t$?
2. Is there any reason to believe there are infinite of them with $t$ not fixed? Is there a conjectural density for such triples with $t$ not fixed?



Density of triple primes
Your questions (more precisely their affirmative answers) are special cases of the generalized Hardy-Littlewood conjecture. You can read about this conjecture in [Linear equations in primes](https://arxiv.org/abs/math/0606088). See especially Conjecture 1.4 on Page 5 and the subsequent remarks on Page 6.


