Primorial

Definition

Let $k$ be a natural number. The $k^{th}$ primorial, sometimes denoted $k\#$, is defined as the product of the first $k$ prime numbers.

The primorial $0\#$ is defined to be $1$.

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 $0,1,2,3,4,5$ are respectively $1,2,6,30,210,2310$.

Special properties

• The primorial $k\#$ is the smallest natural number $n$ with $\omega (n)=k$, where $\omega$ 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.