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]]. | ||
Revision as of 02:40, 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 Failed to parse (lexing error):
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.
