I've been thinking about primorials in the context of the twin prime conjecture. I am seeing this primarily as an exercise to improve my intuition about primorials and prime patterns more than the possibility of making any progress. :-)
So, I thought that it would be interesting to think about numbers of the following form. Let $p_k$ be the $k$th prime. Then, I am interested in the following sequence of numbers:
$\left(p_k\#-1, p_k\#+1\right)$, $\left(\frac{p_k\#}{3}+1,\frac{p_k\#}{3}+3\right)$, $\left(\frac{p_k\#}{3}-1,\frac{p_k\#}{3}-3\right)$, $\left(\frac{p_k\#}{2}+2,\frac{p_k\#}{2}+4\right)$, $\left(\frac{p_k\#}{2}-2,\frac{p_k\#}{2}-4\right)$, $\left(\frac{p_k\#}{15}+3,\frac{p_k\#}{15}+5\right)$, $\left(\frac{p_k\#}{15}-3,\frac{p_k\#}{15}-5\right) \cdots$
Here is my question. Does anyone have any suggestions on methods for counting the number of distinct integeres in this sequence? My basic goal here is to estimate a lower bound for the number of distinct $x$ where $p_k< x \le p_k\#+1$
You asked for a bound on the number of integers between $p_k$ and $p_k\#+1$. $\log x\#\sim x$ and so the number is roughly $e^{p_k}\approx k^k.$