Almost prime

Almost prime

In number theory, a natural number is called "k"-almost prime if and only if it has exactly "k" prime factors, counted with multiplicity. 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":

:Omega(n) := sum a_i qquadmbox{if}qquad n = prod p_i^{a_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 at most "k" prime divisors (not necessarily distinct) is asymptotic tocite book
last = Tenenbaum
first = Gerald
authorlink = Gerald Tenenbaum
coauthors =
title = Introduction to Analytic and Probabilistic Number Theory
publisher = Cambridge University Press
date = 1995
isbn = 0521412617

: pi_k(n) sim left( frac{n}{log n} ight) frac{(loglog n)^{k-1{(k - 1)!},a result of Landau. See also the Hardy–Ramanujan theorem.

References

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