Almost prime ($\Omega (n)$)

An almost prime is a positive integer with few prime factors, counted with multiplicity. WriteA common convention calls $n$ a $k$-almost prime when $\Omega (n)=k$; sieve notation $P_r$ often instead means $\Omega (n)\le r$. The convention must therefore be stated. For example, $12=2^2\cdot 3$ has three prime factors with multiplicity, whereas $15=3\cdot 5$ has two. The number $1$ has $\Omega (1)=0$.

This is a natural output of sieve theory. If $1<n\le X$ and every prime factor of $n$ is at least $z>1$, then $n\ge z^{\Omega (n)}$, hence $\Omega (n)\le \log X/\log z$. Removing sufficiently small prime factors therefore produces almost primes even when proving primality is out of reach. The parity problem in sieve theory explains a major obstruction to upgrading such results to prime values. ^[ref]