Almost prime

In number theory, a natural number is called Template:Mvar-almost prime if it has Template:Mvar prime factors.^[1]^[2]^[3] More formally, a number Template:Mvar is Template:Mvar-almost prime if and only if Template:Math, where Template:Math is the total number of primes in the prime factorization of Template:Mvar (can be also seen as the sum of all the primes' exponents):

Ω (n) : = \sum a_{i} if 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 Template:Mvar-almost primes is usually denoted by Template:Math. The smallest Template:Mvar-almost prime is Template:Math. The first few Template:Mvar-almost primes are:

The number Template:Math of positive integers less than or equal to Template:Mvar with exactly Template:Mvar prime divisors (not necessarily distinct) is asymptotic to:^[4]

π_{k} (n) \sim (\frac{n}{\log n}) \frac{(\log \log n)^{k - 1}}{(k - 1)!},

Properties

The product of a Template:Math-almost prime and a Template:Math-almost prime is a Template:Math-almost prime.
A Template:Mvar-almost prime cannot have a Template:Mvar-almost prime as a factor for all Template:Math.