@YaakovBaruch Thanks for the comment. I will say that with the spreadsheet data, the growth of $G_n << n^2$.

When I was working with the tags, none were allowing logarithmic integral, logarithmic, and integral, so I am sorry about the tags. But, the question comes from $\pi(x * x^\theta) - \pi(x) \ge 1 \text{ for }x \ge 2\text{ with }\theta = \frac{1}{\pi(x)}$ and the note. I am not in the position (no program like Mathematica) to test this "conjecture" out that it is similar to Firoozbakht's conjecture. With that being said, I have not done the same with $x/\log x$ or x/(\log x-1).

One more thing with $p_i$, the is any integer value $i > k$.

$p_k = R_n$ is the nth Ramanujan prime.

251 is the 54th prime, so this 51-tuple will have most of the primes included. This seems similar to A165959 at the OEIS. This sequence does not fix k, but fixes the number of primes in the interval $[p_{i-n}, p_k]$ to n and allows one to find the next prime $p_i < 2*p_{i-n}$. What I am really wondering is if there is a way to combine these two ideas? It would be really cool if someone could prove that there are an infinite number of 3 in the sequence because this would prove the twin prime conjecture. The prime $p_i$ is then next prime after $R_n$ and $p_{i-n}$ is the next one after $R_n/2$.

The difference of the pair $k$ happens for an infinite number of other pairs > 252 with a difference of $k$?

So using the improved numbers of 252 and 51, you are saying that any number from 2 to 51 that do no complete a residue system modulo, which mean that the dividing number is greater than $\ceilimg{\sqrt{51}} = 8$ has a pair with another prime < 252?

OK, so Zhang proof does not touch or cover the values greater than 70 million, but the Pintz paper does. I guess with the current reduction to 252 it is the Pintz paper doing the work for 252 to 70 million.

See COMMENTS by John W. Nicholson, on Dec 14 2013 at oeis.org/A168421 .