Almost prime

In number theory, a natural number is called k-almost prime if it has k prime factors. More formally, a number n is k-almost prime if and only if , where Ω(n) is the total number of primes in the prime factorization of n (can be also seen as the sum of all the primes' exponents):

:\Omega(n) := \sum a_i \qquad\mbox{if}\qquad n = \prod p_i^{a_i}.

A natural number is thus prime if and only if it is 1-almost prime, and semiprime if and only if it is 2-almost prime. The set of k-almost primes is usually denoted by P**k. The smallest k-almost prime is 2k. The first few k-almost primes are:

The number π**k(n) of positive integers less than or equal to n with exactly k prime divisors (not necessarily distinct) is asymptotic to:{{cite book |author-link=Gerald Tenenbaum

: \pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!}, a result of Landau.{{cite book |author-link=Edmund Landau |orig-year=first published 1909