Primorial: Difference between revisions
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==Definition== | ==Definition== | ||
Let <math>k</math> be a [[natural number]]. The <math>k^{th}</math> '''primorial''', sometimes denoted <math>k\ | 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>. | ||
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==Special properties== | ==Special properties== | ||
* The primorial <math>k\ | * 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 [[ | * Every primorial is a [[minimum-so-far]] for the ratio of the [[Euler phi-function]] and the [[identity 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 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.