Frequently, the gap between twice a pair of twin primes contains a prime number. That is, for $p_i,(p_i+2)\in \mathbb P$, it is often but not always the case that one of $2p_i+1$ or $2p_i+3$ is also prime. In fact, for $p_i=29$ both $59,61$ (another pair of twin primes) are prime.
I looked at the first sixty pairs of twin primes (by hand; I'm not a programmer). What I observed is that the number of instances in which a gap between twice a pair of twin primes fails to feature a prime tends to increase as the size of the twin primes increases, but the effect is not particularly rapid. $p_i$ for which $(p_i+2)\in \mathbb P$ and neither $2p_i+1$ nor $2p_i+3$ are also prime include: $71,103,107,109,149,311,347,461,521,569,821,857,881,1061,1091,1151,1301,1319,1487,1619,1667,1697,1721,1787,1871,1877,1949$
Minor question: Are there other instances where $p_i,(p_i+2)\in \mathbb P$, and both $2p_i+1$ and $2p_i+3$ are also prime?
Refinements to Bertrand's Postulate suggest that for arbitrarily small $\epsilon$, there is always a value $n_0$ such that for $n>n_0$ there is a prime $p$ in the gap $n<p<(1+\epsilon)n$. Various formulations of $\epsilon$ with regard to various values of $n_0$ have been advanced. In the case I am looking at, $\epsilon = \frac{2}{p_i}$ which gets arbitrarily small as $p_i$ gets large. Depending on how rapidly the value of $\epsilon$ in my scenario diminishes compared to other evaluations of $\epsilon$, it might become the case that the gaps I am discussing either must always or may never contain a prime number. With regard to the 'must always' option, the data at small values of $p_i$ run counter to it in that primeless gaps occur and appear to increase in frequency as $p_i$ increases.
Main question: Is there a number $n_0$, such that $p_i>n_0$, $(p_i+2)\in \mathbb P$, and one of $2p_i+1$ or $2p_i+3$ must be prime?