Definition:Primorial

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Definition

There are two definitions for primorials: one for primes and one for positive integers.


Definition for Primes

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

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

$\ds p_n \# := \prod_{k \mathop = 1}^n p_k$

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


Definition for Positive Integers

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 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