Editing Primorial (section)

== Definition for natural numbers ==
[[Image:Primorial n plot.png|thumb|300px|{{math|''n''!}} (yellow) as a function of {{math|''n''}}, compared to {{math|''n''#}}(red), both plotted logarithmically.]]

In general, for a positive integer {{mvar|n}}, its primorial, {{math|''n#''}}, is the product of the primes that are not greater than {{mvar|n}}; that is,<ref name="mathworld" /><ref name="OEIS A034386">{{OEIS|id=A034386}}</ref>

:<math>n\# = \prod_{p \le n\atop p \text{ prime}} p = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\# </math>,

where {{math|''π''(''n'')}} is the [[prime-counting function]] {{OEIS|id=A000720}}, which gives the number of primes ≤ {{mvar|n}}. This is equivalent to:

:<math>n\# = 
\begin{cases}
    1 & \text{if }n = 0,\ 1 \\
    (n-1)\# \times n & \text{if } n \text{ is prime} \\
    (n-1)\# & \text{if } n \text{ is composite}.
\end{cases}</math>

For example, 12# represents the product of those primes ≤ 12:

:<math>12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.</math>

Since {{math|''π''(12) {{=}} 5}}, this can be calculated as:

:<math>12\# = p_{\pi(12)}\# = p_5\# = 2310.</math>

Consider the first 12 values of {{math|''n''#}}:

:1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.

We see that for composite {{mvar|n}} every term {{math|''n''#}} simply duplicates the preceding term {{math|(''n'' − 1)#}}, as given in the definition. In the above example we have {{math|12# {{=}} ''p''<sub>5</sub># {{=}} 11#}} since 12 is a composite number.

Primorials are related to the first [[Chebyshev function]], written {{not a typo|{{math|''{{not a typo|ϑ}}''(''n'')}} or {{math|''θ''(''n'')}}}} according to:

:<math>\ln (n\#) = \vartheta(n).</math><ref>{{Mathworld | urlname=ChebyshevFunctions | title=Chebyshev Functions}}</ref>

Since {{math|''{{not a typo|ϑ}}''(''n'')}} asymptotically approaches {{math|''n''}} for large values of {{math|''n''}}, primorials therefore grow according to:
:<math>n\# = e^{(1+o(1))n}.</math>

The idea of multiplying all known primes occurs in some proofs of the [[infinitude of the prime numbers]], where it is used to derive the existence of another prime.