Almost prime

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

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 -almost primes is usually denoted by . The smallest -almost prime is . The first few -almost primes are:

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

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

Properties

• The product of a -almost prime and a -almost prime is a -almost prime.
• A -almost prime cannot have a -almost prime as a factor for all .

References

See also

• sphenic number – name for square-free 3-almost primes

External links