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):

:$\Omega(n) := \sum a_i \qquad\mbox\qquad n = \prod p_i^.$

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 -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:

: $\pi_k(n) \sim \left( \frac \right) \frac,$
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

External links

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Integer sequences
Prime numbers