This site is supported by donations to The OEIS Foundation.

Primorial

From OeisWiki

Jump to: navigation, search

This article needs more work.

Please help by expanding it!


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 \,.

Contents

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!) \,

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

Personal tools