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 $n$th primorial number and the primorial of a natural number $n$.

Primorial numbers

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

$p_n\mathrm{\#}:=\prod _{i=1}^n{p}_i,n\ge 0,$

where $p_i$ is the $i$th prime.

A002110 The primorial numbers, $p_n\mathrm{\#},n\ge 0.$

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

$n\mathrm{\#}:=p_{\pi (n)}\mathrm{\#}=\prod _{i=1}^n{i}^{{\chi }_{\{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{\}}}(i)}=\frac{n!}{\prod _{i=1}^n{i}^{{\chi }_{\{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{\}}}(i)}}=\frac{n!}{\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}(n)},n\ge 0,$

where $\pi (n)$ is the prime counting function, ${\chi }_{\{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{\}}}(i)$ and ${\chi }_{\{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{\}}}(i)$ are the characteristic function of the primes and characteristic function of the composites respectively, $n!$ is the factorial of $n$ and $n\mathrm{\#}$ is the primorial of $n$.

A034386 The primorial of $n$, i.e. $n\mathrm{\#},n\ge 0.$

{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 $n$ is the squarefree kernel $\mathrm{s}\mathrm{q}\mathrm{f}(n!)$, or radical $\mathrm{r}\mathrm{a}\mathrm{d}(n!)$, of $n!$

$n\mathrm{\#}=\mathrm{r}\mathrm{a}\mathrm{d}(n!)$

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