# 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#</math> is defined to be <math>1</math>. | + | 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]]. |

## Revision as of 02:41, 29 April 2009

## 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.

## Behavior

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A002110

### Initial values

The values of primorials at are respectively .

## 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.