$$h(n) = \#\{ \pi(x)\pi(n-x),x\le n\}$$

**What is the growth rate of $h(n)?$**

(the notation means find the distinct values of $h(n)$ for each $n \in \Bbb N)$

for example, plotting the point $(12,4)$ corresponds to $n=12$ and $4$ distinct values for $n=12.$

If the answerer(s) could include plot(s) in their answer with many more values that would be great. I'm interested in seeing a better plot and learning about a possible pattern.

Prime counting function is in light green for reference

Updated plot:

It seems to grow about the same as $\pi(x)$ for this sample set of points, which makes sense because $f(x)$ is defined using the multiplication of prime counting functions. It seems to be less than or equal to $\pi(x)$ for all values.