Innumber theory, anatural number is calledk-almost prime if it haskprime factors.^[1][2][3] More formally, a numbern isk-almost prime if and only ifΩ(n) =k, whereΩ(n) is the total number of primes in theprime factorization ofn (can be also seen as the sum of all the primes' exponents):

${\displaystyle \mathrm{\Omega }(n):=\sum a_i\text{if}n=\prod p_i^{a_i}.}$

A natural number is thusprime if and only if it is 1-almost prime, andsemiprime if and only if it is 2-almost prime. The set ofk-almost primes is usually denoted byP_k. The smallestk-almost prime is2^k. The first fewk-almost primes are:

The numberπ_k(n) of positive integers less than or equal ton with exactlyk prime divisors (not necessarily distinct) isasymptotic to:^[4]

${\displaystyle {\pi }_k(n)\sim \left(\frac{n}{\log n}\right)\frac{(\log \log n{)}^{k-1}}{(k-1)!},}$

a result ofLandau.^[5] See also theHardy–Ramanujan theorem.^[relevant?]

• The product of ak₁-almost prime and ak₂-almost prime is a(k₁ +k₂)-almost prime.
• Ak-almost prime cannot have an-almost prime as a factor for alln >k.

• Weisstein, Eric W."Almost prime".MathWorld.

• Integer sequences
• Prime numbers

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