Let's define our "prime sequence analogue"
$\{a_n\}\subset \Bbb N$ and satisfies some properties that I am trying to work out
A sequence that satisfies these unkown properties is the primes
one property could be stated as: There should be a sieve for generation of all $a_n$ below some integer $k$. For the primes this is the sieve of Eratosthenes and we have $\pi(k)$
Now I am guessing educatedly here. A PNT style asymptotic relation for the counting function of $a_n$.
I think there was a result somewhere which stated roughly that most $a_n \sim A k \log k$
