Definition:Primorial

Definition

There are two definitions for primorials, one for primes and one for natural numbers.

Definition for Primes

Let $p_n$ be the $n$th prime number.

Then the $n$th primorial $p_n \#$ is defined as:

$\displaystyle p_n \# := \prod_{i=1}^n p_k$

That is, $p_n \#$ is the product of the first $n$ primes.

Definition for Natural Numbers

Let $n$ be a natural number.

Then:

$\displaystyle n\# := \prod_{i=1}^{\pi \left({n}\right)} p_i = p_{\pi \left({n}\right)}\#$

That is, $n\#$ is defined as the product of all primes less than or equal to $n$.

Thus:

$n\# = \begin{cases} 0 & : n \le 1 \\ n \left({\left({n-1}\right)\#}\right) & : n \mbox { prime} \\ \left({n-1}\right)\# & : n \mbox { composite} \end{cases}$

Examples

The first few primorials (of both types) are as follows:

	\(\displaystyle \)	\(\displaystyle 1\#\)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle = 0\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle 2\#\)	\(\displaystyle \)	\(\displaystyle = p_1 \#\)	\(\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle = 2\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle 3\#\)	\(\displaystyle \)	\(\displaystyle = p_2\#\)	\(=\)	\(\displaystyle 2 \times 3\)	\(\displaystyle \)	\(\displaystyle = 6\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle 4\#\)	\(\displaystyle \)	\(\displaystyle = p_2\#\)	\(=\)	\(\displaystyle 2 \times 3\)	\(\displaystyle \)	\(\displaystyle = 6\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle 5\#\)	\(\displaystyle \)	\(\displaystyle = p_3\#\)	\(=\)	\(\displaystyle 2 \times 3 \times 5\)	\(\displaystyle \)	\(\displaystyle = 30\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle 6\#\)	\(\displaystyle \)	\(\displaystyle = p_3\#\)	\(=\)	\(\displaystyle 2 \times 3 \times 5\)	\(\displaystyle \)	\(\displaystyle = 30\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle 7\#\)	\(\displaystyle \)	\(\displaystyle = p_4\#\)	\(=\)	\(\displaystyle 2 \times 3 \times 5 \times 7\)	\(\displaystyle \)	\(\displaystyle = 210\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle 8\#\)	\(\displaystyle \)	\(\displaystyle = p_4\#\)	\(=\)	\(\displaystyle 2 \times 3 \times 5 \times 7\)	\(\displaystyle \)	\(\displaystyle = 210\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle 9\#\)	\(\displaystyle \)	\(\displaystyle = p_4\#\)	\(=\)	\(\displaystyle 2 \times 3 \times 5 \times 7\)	\(\displaystyle \)	\(\displaystyle = 210\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle 10\#\)	\(\displaystyle \)	\(\displaystyle = p_4\#\)	\(=\)	\(\displaystyle 2 \times 3 \times 5 \times 7\)	\(\displaystyle \)	\(\displaystyle = 210\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle 11\#\)	\(\displaystyle \)	\(\displaystyle = p_5\#\)	\(=\)	\(\displaystyle 2 \times 3 \times 5 \times 7 \times 11\)	\(\displaystyle \)	\(\displaystyle = 2310\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle 12\#\)	\(\displaystyle \)	\(\displaystyle = p_5\#\)	\(=\)	\(\displaystyle 2 \times 3 \times 5 \times 7 \times 11\)	\(\displaystyle \)	\(\displaystyle = 2310\)	\(\displaystyle \)

This sequence is A002110 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).