Primorial - Revision history

Vipul: /* Behavior */

2012-06-23T01:04:14Z

Behavior

← Older revision		Revision as of 01:04, 23 June 2012
Line 7:		Line 7:
	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~~==		==Occurrence==

	~~{{oeis\|A002110}}~~

	===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~~>.		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==

Vipul at 02:41, 29 April 2009

2009-04-29T02:41:03Z

← Older revision		Revision as of 02:41, 29 April 2009
Line 3:		Line 3:
	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]].

Vipul at 02:40, 29 April 2009

2009-04-29T02:40:52Z

← Older revision		Revision as of 02:40, 29 April 2009
Line 3:		Line 3:
	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>.		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]].

Vipul at 02:40, 29 April 2009

2009-04-29T02:40:34Z

← Older revision		Revision as of 02:40, 29 April 2009
Line 1:		Line 1:
	==Definition==		==Definition==

	Let <math>k</math> be a [[natural number]]. The <math>k^{th}</math> '''primorial''', sometimes denoted <math>k\~~sharp~~</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>.		The primorial <math>0\sharp</math> is defined to be <math>1</math>.
Line 17:		Line 17:
	==Special properties==		==Special properties==

	* The primorial <math>k\~~sharp~~</math> is the smallest natural number <math>n</math> with <math>\omega(n) = k</math>, where <math>\omega</math> is the [[prime divisor count function]].		* The primorial <math>k\#</math> is the smallest natural number <math>n</math> with <math>\omega(n) = k</math>, where <math>\omega</math> is the [[prime divisor count function]].
	* Every primorial is a [[~~record~~ minimum]] for the ratio of the [[Euler phi-function]] and the [[identity function]].		* Every primorial is a [[minimum-so-far]] for the ratio of the [[Euler phi-function]] and the [[identity function]].

Vipul: Created page with '==Definition== Let $k$ be a natural number. The $k^{th}$ '''primorial''', sometimes denoted $k\sharp$ , is defined as the product of the firs...'

2009-04-29T02:39:32Z

Created page with '==Definition== Let <math>k</math> be a natural number. The <math>k^{th}</math> '''primorial''', sometimes denoted <math>k\sharp</math>, is defined as the product of the firs...'

New page

==Definition==

Let <math>k</math> be a [[natural number]]. The <math>k^{th}</math> '''primorial''', sometimes denoted <math>k\sharp</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>.

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==

{{oeis|A002110}}

===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>.

==Special properties==

* The primorial <math>k\sharp</math> is the smallest natural number <math>n</math> with <math>\omega(n) = k</math>, where <math>\omega</math> is the [[prime divisor count function]].
* Every primorial is a [[record minimum]] for the ratio of the [[Euler phi-function]] and the [[identity function]].

Primorial - Revision history

Vipul: /* Behavior */

Vipul at 02:41, 29 April 2009

Vipul at 02:40, 29 April 2009

Vipul at 02:40, 29 April 2009

Vipul: Created page with '==Definition== Let k be a natural number. The k^{th} '''primorial''', sometimes denoted k\sharp, is defined as the product of the firs...'

Vipul: Created page with '==Definition== Let $k$ be a natural number. The $k^{th}$ '''primorial''', sometimes denoted $k\sharp$ , is defined as the product of the firs...'