Definition:Primorial/Positive Integer

Definition

Let $n$ be a positive integer.

Then:

$\ds n\# := \prod_{i \mathop = 1}^{\map \pi n} p_i = p_{\map \pi n}\#$

where $\map \pi n$ is the prime counting function.

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

Thus:

$n\# = \begin {cases}

1 & : n \le 1 \\ n \paren {\paren {n - 1}\#} & : \text {$n$ prime} \\ \paren {n - 1}\# & : \text {$n$ composite} \end {cases}$

Examples

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

\(\ds 0\# \ \ \)

\(\ds = p_0 \#\)

\(=\)

\(\ds \)

\(\ds = 1\)

\(\ds 1\# \ \ \)

\(\ds = p_0 \#\)

\(=\)

\(\ds \)

\(\ds = 1\)

\(\ds 2\# \ \ \)

\(\ds = p_1 \#\)

\(=\)

\(\ds \)

\(\ds = 2\)

\(\ds 3\# \ \ \)

\(\ds = p_2\#\)

\(=\)

\(\ds 2 \times 3\)

\(\ds = 6\)

\(\ds 4\# \ \ \)

\(\ds = p_2\#\)

\(=\)

\(\ds 2 \times 3\)

\(\ds = 6\)

\(\ds 5\# \ \ \)

\(\ds = p_3\#\)

\(=\)

\(\ds 2 \times 3 \times 5\)

\(\ds = 30\)

\(\ds 6\# \ \ \)

\(\ds = p_3\#\)

\(=\)

\(\ds 2 \times 3 \times 5\)

\(\ds = 30\)

\(\ds 7\# \ \ \)

\(\ds = p_4\#\)

\(=\)

\(\ds 2 \times 3 \times 5 \times 7\)

\(\ds = 210\)

\(\ds 8\# \ \ \)

\(\ds = p_4\#\)

\(=\)

\(\ds 2 \times 3 \times 5 \times 7\)

\(\ds = 210\)

\(\ds 9\# \ \ \)

\(\ds = p_4\#\)

\(=\)

\(\ds 2 \times 3 \times 5 \times 7\)

\(\ds = 210\)

\(\ds 10\# \ \ \)

\(\ds = p_4\#\)

\(=\)

\(\ds 2 \times 3 \times 5 \times 7\)

\(\ds = 210\)

\(\ds 11\# \ \ \)

\(\ds = p_5\#\)

\(=\)

\(\ds 2 \times 3 \times 5 \times 7 \times 11\)

\(\ds = 2310\)

\(\ds 12\# \ \ \)

\(\ds = p_5\#\)

\(=\)

\(\ds 2 \times 3 \times 5 \times 7 \times 11\)

\(\ds = 2310\)

\(\ds 13\# \ \ \)

\(\ds = p_6\#\)

\(=\)

\(\ds 2 \times 3 \times 5 \times 7 \times 11 \times 13\)

\(\ds = 30 \, 030\)

The sequence contines:

$1, 2, 6, 30, 210, 2310, 30 \, 030, 510 \, 510, 9 \, 699 \, 690, 223 \, 092 \, 870, \ldots$

Sources

• Weisstein, Eric W. "Primorial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Primorial.html