Definition:Primorial

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:


 * $$p_n \# \ \stackrel {\mathbf {def}} {=\!=} \ \prod_{i=1}^n p_k$$

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

Definition for Natural Numbers
Let $$n$$ be a natural number.

Then $$n\# \ \stackrel {\mathbf {def}} {=\!=} \ \prod_{i=1}^{\pi \left({n}\right)} p_i = p_{\pi \left({n}\right)}\#$$.

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:

$$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$