Between the prime numbers $n=1327$ and $n+k = 1327+34 = 1361$ there are $k-1=33$ consecutive composite numbers.
If you double those bounding primes, getting $2\times1327=2654$ and $2\times1361=2722$, then between them you find $14$ primes, i.e. in the interval $[2654,2722]$ you have $2k+1=69$ numbers of which one out of every $69/14 = 4.92857\ldots$ is prime. With numbers that big you'd expect somewhat fewer than one in six or seven, if I'm not mistaken.
Herewith a wild guess: Is this larger interval compensating for the dearth of primes in that smaller interval, in the sense that for consecutive primes $n$ and $n+k$ with $k/n$ unusually big, you would typically find a more-than-expected number of primes between $2n$ and $2(n+k)$?
PS: For the record: $$2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719$$
