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 Ω(n) = k, 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 Pk. The smallest k-almost prime is 2k. The first few k-almost primes are:



The number &pi;k(n) of positive integers less than or equal to n with exactly k prime divisors (not necessarily distinct) is asymptotic to:


 * $$ \pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!},$$

a result of Landau. See also the Hardy–Ramanujan theorem.

Properties

 * The multiple of a $$k_1$$-almost prime and a $$k_2$$-almost prime is a $$(k_1+k_2)$$-almost prime.
 * A $$k$$-almost prime cannot have a $$n$$-almost prime as a factor for all $$n>k$$.