Say a prime number $p$ is of order $k$ if $k$ is the smallest non negative number $f$ such that the $f$-th iterate of the prime counting function, denoted by $\pi^{(f)}$, verifies $\pi^{(f)}(p)$ is not itself prime. Hence $2$ has order $1$, $3$ has order $2$, $4$ has order $0$, and so on.
Is there a way to take the order of a prime into account in sieve methods to get a better knowledge of the distribution of primes?