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.

Occurrence

Initial values

The values of primorials at $0,1,2,3,4,5,\dots$ are given in the list:

1, 2, 6, 30, 210, 2310 View list on OEIS

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.