# Primorial

From Number

## 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.

## Occurrence

### Initial values

The values of primorials at are given in the list: 1, 2, 6, 30, 210, 2310 View list on OEIS

## 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.