Primorial

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

Primorial numbers

The $\scriptstyle n\,$th primorial number, denoted $\scriptstyle p_{n}\#\,$, is defined as the product of the first $\scriptstyle n\,$ 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 $\scriptstyle p_{i}\,$ is the $\scriptstyle i\,$th prime.

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

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

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

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

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

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

See also

• Factorial
• Compositorial