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Let $p_n$ denote the $n^\text{th}$ prime. Find a lower bound for $\left|S\right|$ where

$$S = \left\{ q \in \mathbb{N} \mid q \text{ is prime and } p_n - n \leq q \leq p_n + n \right\}. $$

Any good bounds known? See this graph: http://oeis.org/A097935/graph

[Edit 1] Let

$$ a = n\left(\ln(n) + \frac{13}{500}\right), \ \ b = n\left(\ln(n) - \frac{987}{500}\right)$$ and $$ r = \frac{a}{\ln(a)-1} - \frac{b}{\ln(b)-23/20} - 2. $$

Conjecture 1: For $n \ge 12$ $$ \lfloor r \rfloor \le |S|. $$

[Edit 2] It seems that we can push things a little further. Let

$$ s = \frac{a}{\ln(a)-1} - \frac{b}{\ln(b)-23/20} + \frac{a}{\ln(a)-23/20} - \frac{b}{\ln(b)-1} $$

Conjecture 2: $s/2$ and $|S|$ are asymptotical equivalent.

$$ \frac{s}{2} \sim |S| $$

For the fun of it let's look at an numerical example. Let $n = 10000$. Then $|S| = 1715$ and $s/2 = 1762.31..$.

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