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 continued 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 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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $6$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6$
- Weisstein, Eric W. "Primorial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Primorial.html