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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 $\scriptstyle n \,$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 0th primorial number being the empty product, i.e. 1)

$p_n\# := \prod_{i=1}^{n} p_i,\quad n \ge 0, \,$

where $\scriptstyle p_i \,$ is the $\scriptstyle i \,$th prime.

A002110 The primorial numbers, $\scriptstyle p_n\#,\ n \,\ge\, 0. \,$

{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)

$n\# := p_{\pi(n)}\# = \prod_{i=1}^{n} i^{ \chi_{\{\rm primes\}}(i) } = \frac{n!}{\prod_{i=1}^{n} i^{ \chi_{\{\rm composites\}} (i) }} = \frac{n!}{{\rm Compositorial}(n)},\quad n \ge 0, \,$

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. $\scriptstyle n\#,\ n \,\ge\, 0. \,$

{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