The sequence in question is: $$S=\{\int_0^1\pi(x)\pi(1-x)dx,\int_0^2\pi(x)\pi(2-x)dx,...\},$$ 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^{...

So $\pi(x)$ is the prime counting function. That is to say, it counts the number of primes below a given integer $x$. This function is very important in number theory. I was wondering how well the following counts primes: $\phi(x)-c(x) = \int_2^x e^{1/\ln(x)}dx-\int_2^x 1 dx .$ I tried making a t...