= Almost prime
{title2=$\Omega(n)$}
An almost prime is a positive <integer> with few <prime factors>, counted with multiplicity. Write
$$
n=\prod_p p^{a_p},\qquad \Omega(n)=\sum_pa_p.
$$
A common convention calls $n$ a $k$-almost prime when $\Omega(n)=k$; sieve notation $P_r$ often instead means $\Omega(n)\leq r$. The convention must therefore be stated. For example, $12=2^2\cdot3$ has three <prime factors> with multiplicity, whereas $15=3\cdot5$ has two. The number $1$ has $\Omega(1)=0$.
This is a natural output of <sieve theory>. If $1<n\leq X$ and every <prime factor> of $n$ is at least $z>1$, then $n\geq z^{\Omega(n)}$, hence $\Omega(n)\leq\log X/\log z$. 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. <Some sieve theory by Terence Tao (254A Notes 4)>[Introduction and applications following Corollary 19]{ref}
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