# Efficiency of the Sieve of Eratosthenes.

It's well-known that the sieve of Eratosthenes, using the first $m$ primes {$p_1, p_2, ..., p_m$}, sifts out all composite numbers up to $(p_m+2)^2$, since every composite $n \lt (p_m+2)^2$ contains (at least) one of the $p_i$ in its factorization and so is caught by the sieve.

A similar observation also gives that the sieve catches all the numbers with at least 3 prime factors up to $(p_m + 2)^3$, those with at least 4 prime factors up to $(p_m + 2)^4$, and so-on.

But in fact it seems to catch even more than this: for example, when we use just $P$ = {2,3} for the sieve, a large portion of the numbers between $(3+2)^2 = 5^2 = 25$ and $5^3 = 125$ with two prime factors happen to get caught as well, but a smaller portion with three prime factors get caught between $125$ and $5^4 = 625$, and an even smaller portion with two prime factors in that range. (I think.) Heuristically, this seems to be just because there are larger prime factors to throw together into the factorization.

Are these statistics known? More precisely,

• What is the probability that the sieve catches a number with $2$ prime factors in the intervals $[(p_m + 2)^2,(p_m + 2)^3]$, $[(p_m + 2)^3,(p_m + 2)^4]$, etc., or with $3$ prime factors in the intervals $[(p_m + 2)^3,(p_m + 2)^4]$, $[(p_m + 2)^4,(p_m + 2)^5]$, etc., etc.

In general, just how efficient is the sieve of Eratosthenes, beyond the "guaranteed" cases described above? Is there an easy way to characterize all this?