Almost prime

In number theory, a natural number is called almost prime if there exists an absolute constant K such that the number has at most K prime factors. An almost prime n is denoted by P_r if and only if the number of prime factors of n, counted according to multiplicity, is at most r. A natural number is called k-almost prime if it has exactly k prime factors, counted with multiplicity. 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:

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 2^k. The first few k-almost primes are:

The number π_k(n) of positive integers less than or equal to n with at most k prime divisors (not necessarily distinct) is asymptotic to:

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

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