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# Primorial

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

## Primorial numbers

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

${\displaystyle p_{n}\#:=\prod _{i=1}^{n}p_{i},\quad n\geq 0,\,}$

where ${\displaystyle \scriptstyle p_{i}\,}$ is the ${\displaystyle \scriptstyle i\,}$th prime.

A002110 The primorial numbers, ${\displaystyle \scriptstyle p_{n}\#,\ n\,\geq \,0.\,}$

{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 ${\displaystyle \scriptstyle n\,}$ (the primorial of ${\displaystyle \scriptstyle n\,}$), denoted ${\displaystyle \scriptstyle n\#\,}$, is the product of all primes up to ${\displaystyle \scriptstyle n\,}$ (the primorial of 0 being the empty product, i.e. 1)

${\displaystyle n\#:=p_{\pi (n)}\#=\prod _{i=1}^{n}i^{\chi _{\{{\rm {primes\}}}}(i)}={\frac {n!}{\prod _{i=1}^{n}i^{\chi _{\{{\rm {composites\}}}}(i)}}}={\frac {n!}{{\rm {Compositorial}}(n)}},\quad n\geq 0,\,}$

where ${\displaystyle \scriptstyle \pi (n)\,}$ is the prime counting function, ${\displaystyle \scriptstyle \chi _{\{{\rm {primes\}}}}(i)\,}$ and ${\displaystyle \scriptstyle \chi _{\{{\rm {composites\}}}}(i)\,}$ are the characteristic function of the primes and characteristic function of the composites respectively, ${\displaystyle \scriptstyle n!\,}$ is the factorial of ${\displaystyle \scriptstyle n\,}$ and ${\displaystyle \scriptstyle n\#\,}$ is the primorial of ${\displaystyle \scriptstyle n\,}$.

A034386 The primorial of ${\displaystyle \scriptstyle n\,}$, i.e. ${\displaystyle \scriptstyle n\#,\ n\,\geq \,0.\,}$

{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 ${\displaystyle \scriptstyle n\,}$ is the squarefree kernel ${\displaystyle \scriptstyle {\rm {sqf}}(n!)\,}$, or radical ${\displaystyle \scriptstyle {\rm {rad}}(n!)\,}$, of ${\displaystyle \scriptstyle n!\,}$

${\displaystyle n\#={\rm {rad}}(n!)\,}$

## 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