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# 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 ^{th} **primorial number** and the **primorial of a natural number** .

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## Primorial numbers

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

where is the ^{th} prime.

A002110 The primorial numbers,

- {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 (the **primorial** of ), denoted , is the product of all primes up to (the primorial of 0 being the empty product, i.e. 1)

where is the prime counting function, and are the characteristic function of the primes and characteristic function of the composites respectively, is the factorial of and is the primorial of .

A034386 The **primorial** of , i.e.

- {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 is the squarefree kernel , or radical , of

## 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