Primorial

English

Template:Lb Any number belonging to the integer sequence whose nth element is the product of the first n primes.
Template:Synonyms
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Template:Lb A unary operation, denoted by the postfix symbol # and defined on the nonnegative integers, which maps 0 to 1, 1 to 1, and each subsequent number to the product of all primes less than or equal to it; the value mapped to by said operation for a given input.
Template:Ux
- 2020, Rong Pan, Qinheping Hu, Rishabh Singh, Loris D'Antoni, Solving Problem Sketches with Large Integer Values, Peter Müller (editor), Programming Languages and Systems: 29th European Symposium, Proceedings, Springer, Template:W 12075, page 587,
  The following number theory result relates the primorial to the Chebyshev function.
  $ϑ (n) = \log (n #) = \log 2^{(1 + o (n)) n} = (1 + o (n)) n$

The primorial operation may be defined as:

n # = \prod_{\binom{p \leq n}{p 𝗉 𝗋 𝗂 𝗆 𝖾}} p = \prod_{i = 1}^{π (n)} p_{i} = p_{π (n)} #

,

where $π (n)$ denotes the prime-counting function, which gives the number of primes $\leq n$ .

It can also be defined recursively:

n # = {\begin{matrix} 1 & 𝗂 𝖿 n = 0, 1 \\ (n - 1) # \times n & 𝗂 𝖿 n 𝗂 𝗌 𝗉 𝗋 𝗂 𝗆 𝖾 \\ (n - 1) # & 𝗂 𝖿 n 𝗂 𝗌 𝖼 𝗈 𝗆 𝗉 𝗈 𝗌 𝗂 𝗍 𝖾 . \end{matrix}