We know that the probability that a number $n$ is prime is roughly $\frac{1}{\ln(n)}$. My question is, if $A_k$ is the event that $2k-1$ is a prime, are pairs $(A_{k}, A_{k+\Delta k})$ independent? In other words, if a certain number is prime (or not), does that change the likelihood that any surrounding odd numbers are prime? Perhaps more rigorously, as a specific example, is it true that, for almost all $k$, $\Pr[A_k | A_{k-1}] = \Pr[A_k] \sim \frac{1}{\ln(2k+1)}$?
Also, has this been proven/disproven? If not, what does the empirical evidence show?