View source for Almost-prime number

A natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011990/a0119901.png" /> of the form

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where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011990/a0119903.png" /> are prime numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011990/a0119904.png" /> is a constant. Prime numbers are the special case of almost-prime numbers for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011990/a0119905.png" />. There are theorems for almost-prime numbers that generalize theorems on the distribution of prime numbers in the set of natural numbers. Several [[Additive problems|additive problems]] that have not yet been solved for prime numbers have been solved for almost-prime numbers.



====Comments====
See also [[Sieve method|Sieve method]].

An example of a result on the distribution of almost-prime numbers generalizing the corresponding one on prime numbers is the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011990/a0119906.png" /> be the number of square-free almost-prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011990/a0119907.png" />. Then (cf. [[#References|[a1]]], Sect. 22.18)

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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Clarendon Press  (1965)</TD></TR></table>