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Almost prime

Concept in number theory related to prime numbers From Wikipedia, the free encyclopedia

Almost prime
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In number theory, a natural number is called k-almost prime if it has k prime factors.[1][2][3] 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):

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Demonstration, with Cuisenaire rods, of the 2-almost prime nature of the number 6

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:

More information k, OEIS sequence ...

The number πk(n) of positive integers less than or equal to n with exactly k prime divisors (not necessarily distinct) is asymptotic to:[4]

a result of Landau.[5] See also the Hardy–Ramanujan theorem.[relevant?]

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Properties

  • The product of a k1-almost prime and a k2-almost prime is a (k1 + k2)-almost prime.
  • A k-almost prime cannot have a n-almost prime as a factor for all n > k.

References

See also

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