The sequence in question is:
$$S=\left\{\int_0^1\pi(x)\pi(1-x)dx,\int_0^2\pi(x)\pi(2-x)dx,...\right\},$$
where $\pi(x)$ is the prime counting function.
I don't know how to check this for an infinite sequence but I've tried computing many values.
Here's the first prime number in the sequence:
$$\int_0^{13}\pi(x)\pi(13-x)dx=73.$$
and the second in the sequence: $$\int_0^{57}\pi(x)\pi(57-x)dx=3803.$$
This is what I know: The primes thin out as higher numbers are reached. My conclusion is that this sequence will continue to find fewer primes compared to all values computed.