I am wondering if there is an elementary sequence that has the same density as the prime numbers.
I tried building one by construction by solving when $n = \frac{x}{\ln (x)}$ is an integer, $n \geq 3$. I got the sequence: $e^{-W_{-1}\frac{1}{n}}$ with $W$ being the product logarithm.
Unfortunately, this isn't elementary and it always outputs a transcendental number. Though, it is much easier to compute than the finding the consecutive primes.