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:

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

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


Definition for Positive Integers

Let $n$ be a positive integer.

Then:

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

where $\pi \left({n}\right)$ is the prime counting function.


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

Thus:

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


Examples

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

\(\displaystyle 0\# \ \ \) \(\displaystyle = p_0 \#\) \(=\) \(\displaystyle \) \(\displaystyle = 1\)
\(\displaystyle 1\# \ \ \) \(\displaystyle = p_0 \#\) \(=\) \(\displaystyle \) \(\displaystyle = 1\)
\(\displaystyle 2\# \ \ \) \(\displaystyle = p_1 \#\) \(=\) \(\displaystyle \) \(\displaystyle = 2\)
\(\displaystyle 3\# \ \ \) \(\displaystyle = p_2\#\) \(=\) \(\displaystyle 2 \times 3\) \(\displaystyle = 6\)
\(\displaystyle 4\# \ \ \) \(\displaystyle = p_2\#\) \(=\) \(\displaystyle 2 \times 3\) \(\displaystyle = 6\)
\(\displaystyle 5\# \ \ \) \(\displaystyle = p_3\#\) \(=\) \(\displaystyle 2 \times 3 \times 5\) \(\displaystyle = 30\)
\(\displaystyle 6\# \ \ \) \(\displaystyle = p_3\#\) \(=\) \(\displaystyle 2 \times 3 \times 5\) \(\displaystyle = 30\)
\(\displaystyle 7\# \ \ \) \(\displaystyle = p_4\#\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7\) \(\displaystyle = 210\)
\(\displaystyle 8\# \ \ \) \(\displaystyle = p_4\#\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7\) \(\displaystyle = 210\)
\(\displaystyle 9\# \ \ \) \(\displaystyle = p_4\#\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7\) \(\displaystyle = 210\)
\(\displaystyle 10\# \ \ \) \(\displaystyle = p_4\#\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7\) \(\displaystyle = 210\)
\(\displaystyle 11\# \ \ \) \(\displaystyle = p_5\#\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7 \times 11\) \(\displaystyle = 2310\)
\(\displaystyle 12\# \ \ \) \(\displaystyle = p_5\#\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7 \times 11\) \(\displaystyle = 2310\)
\(\displaystyle 13\# \ \ \) \(\displaystyle = p_6\#\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13\) \(\displaystyle = 30 \, 030\)

The sequence contines:

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

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


Sources