# Difference between revisions of "Primorial"

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Let <math>k</math> be a [[natural number]]. The <math>k^{th}</math> '''primorial''', sometimes denoted <math>k\#</math>, is defined as the product of the first <math>k</math> [[prime number]]s. | Let <math>k</math> be a [[natural number]]. The <math>k^{th}</math> '''primorial''', sometimes denoted <math>k\#</math>, is defined as the product of the first <math>k</math> [[prime number]]s. | ||

− | The primorial <math>0\ | + | The primorial <math>0\#</math> is defined to be <math>1</math>. |

An alternate definition of primorial, called here the [[primorial of the second kind]], is defined as the product of all the primes less than or equal to a given number. Note that the logarithm of the primorial of the second kind is the [[first Chebyshev function]]. | An alternate definition of primorial, called here the [[primorial of the second kind]], is defined as the product of all the primes less than or equal to a given number. Note that the logarithm of the primorial of the second kind is the [[first Chebyshev function]]. | ||

− | == | + | ==Occurrence== |

− | + | ||

− | + | ||

===Initial values=== | ===Initial values=== | ||

− | The values of primorials at <math>0,1,2,3,4,5</math> are | + | The values of primorials at <math>0,1,2,3,4,5,\dots</math> are given in the list: <section begin="list"/>[[1]], [[2]], [[6]], [[30]], [[210]], [[2310]] [[Oeis:A002110|View list on OEIS]]<section end="list"/> |

==Special properties== | ==Special properties== |

## Latest revision as of 01:04, 23 June 2012

## Definition

Let be a natural number. The **primorial**, sometimes denoted , is defined as the product of the first prime numbers.

The primorial is defined to be .

An alternate definition of primorial, called here the primorial of the second kind, is defined as the product of all the primes less than or equal to a given number. Note that the logarithm of the primorial of the second kind is the first Chebyshev function.

## Occurrence

### Initial values

The values of primorials at are given in the list: 1, 2, 6, 30, 210, 2310 View list on OEIS

## Special properties

- The primorial is the smallest natural number with , where is the prime divisor count function.
- Every primorial is a minimum-so-far for the ratio of the Euler phi-function and the identity function.