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# Primorial

(Redirected from Primorial numbers)

A primorial is a product of consecutive prime numbers, starting with the first prime, namely 2. One distinguishes between the $n\,$ th primorial number and the primorial of a natural number $n\,$ .

## Primorial numbers

The $n\,$ th primorial number, denoted $p_{n}\#\,$ , is defined as the product of the first $n\,$ primes (the 0 th primorial number being the empty product, i.e. 1)

$p_{n}\#:=\prod _{i=1}^{n}p_{i},\quad n\geq 0,\,$ where $p_{i}\,$ is the $i\,$ th prime.

A002110 The primorial numbers, $p_{n}\#,\ n\,\geq \,0.\,$ {1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, ...}

## Primorial of natural numbers

The primorial of a natural number $n\,$ (the primorial of $n\,$ ), denoted $n\#\,$ , is the product of all primes up to $n\,$ (the primorial of 0 being the empty product, i.e. 1)

$n\#:=p_{\pi (n)}\#=\prod _{i=1}^{n}i^{\chi _{\{{\rm {primes\}}}}(i)}={\frac {n!}{\prod _{i=1}^{n}i^{\chi _{\{{\rm {composites\}}}}(i)}}}={\frac {n!}{{\rm {Compositorial}}(n)}},\quad n\geq 0,\,$ where $\pi (n)\,$ is the prime counting function, $\chi _{\{{\rm {primes\}}}}(i)\,$ and $\chi _{\{{\rm {composites\}}}}(i)\,$ are the characteristic function of the primes and characteristic function of the composites respectively, $n!\,$ is the factorial of $n\,$ and $n\#\,$ is the primorial of $n\,$ .

A034386 The primorial of $n\,$ , i.e. $n\#,\ n\,\geq \,0.\,$ {1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310, 30030, 30030, 30030, 30030, 510510, 510510, 9699690, 9699690, 9699690, 9699690, 223092870, 223092870, 223092870, 223092870, 223092870, ...}

The primorial of $n\,$ is the squarefree kernel ${\rm {sqf}}(n!)\,$ , or radical ${\rm {rad}}(n!)\,$ , of $n!\,$ $n\#={\rm {rad}}(n!)\,$ ## Product of consecutive primes

The quotient of two primorial numbers gives a product of consecutive primes.

## Sequences

A129912 Numbers that are products of distinct primorial numbers (primorial numbers being a subset). (Related to odd primes distribution conjecture.)

{1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, 2310, 2520, 4620, 6300, 12600, 13860, 27720, 30030, 37800, 60060, 69300, 75600, 138600, 180180, 360360, 415800, 485100, ...}

Conjecture: every odd prime number must either be adjacent to or a prime distance away [i.e. a noncomposite distance away] from a primorial or primorial product (the distance will be a prime smaller than the candidate). - Bill McEachen, Jun 03 2010