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\sharp</math> is defined to be <math>1</math>.
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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]].
  
==Behavior==
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==Occurrence==
 
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{{oeis|A002110}}
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===Initial values===
 
===Initial values===
  
The values of primorials at <math>0,1,2,3,4,5</math> are respectively <math>1,2,6,30,210,2310</math>.
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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