Prime Counting Function Difference If $\pi\left(x\right)$ is the prime counting function, is anything known about the number of x values satisfy 
$$\pi\left(2x\right)-\pi\left(x\right)=1$$
Sorry if this is a stupid question, but I am just curious about the size of such $x$ values and perhaps some properties that they may have. Thanks.
As you know, Bertrand's postulate states that $\pi(2x) - \pi(x) \ge 1$ for all $x \ge 2$. A refinement due to Nagura in 1952 states that as long as $x \ge 25$ then $\pi(6x/5) - \pi(x) \ge 1$. But then $\pi(36x/25) - \pi(6x/5) \ge 1$ too, and since $36/25 &lt; 2$ leads to $\pi(2x) - \pi(x) \ge 2$. This can be improved significantly, but as far as your question goes, the values of $x$ satisfying $\pi(2x) - \pi(x) = 1$ are very limited in number.
The prime counting function is defined for real numbers $x$.  The exact set of real numbers $x$ for which $\pi(2x) - \pi(x) = 1$ is the union of the following intervals:

$$[1,1.5) \cup [2,2.5) \cup [3,3.5) \cup [5,5.5).$$

You can see a sketch of this in the graph (Wolfram Alpha).