Almost Prime

In number theory, a natural number is called ''k''-almost prime if it has ''k'' prime factors. 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):

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

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 ''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 exactly ''k'' 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 multiple of a $k_1$-almost prime and a $k_2$-almost prime is a $(k_1+k_2)$-almost prime.
*A $k$-almost prime cannot have a $n$-almost prime as a factor for all $n>k$.

References

External links

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Integer sequences
Prime numbers