primorial

Etymology
. Coined by American engineer and mathematician.

Noun

 * 1)  Any number belonging to the integer sequence whose nth element is the product of the first n primes.
 * 2)  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.
 * 3) * 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, 12075, page 587,
 * The following number theory result relates the primorial to the Chebyshev function.
 * $$\vartheta(n) = \log(n\#) = \log{2^{(1+o(n))n}} = (1+o(n))n$$
 * 1) * 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, 12075, page 587,
 * The following number theory result relates the primorial to the Chebyshev function.
 * $$\vartheta(n) = \log(n\#) = \log{2^{(1+o(n))n}} = (1+o(n))n$$

Usage notes
The primorial operation may be defined as:
 * $$n\# = \prod_{p \le n\atop p~\mathsf{prime}} p = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\# $$,

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

It can also be defined recursively:
 * $$n\# =

\begin{cases} 1 & \mathsf{if}~n = 0,\ 1 \\ (n-1)\# \times n & \mathsf{if}~n~\mathsf{is~prime} \\ (n-1)\# & \mathsf{if }~n~\mathsf{is~composite}. \end{cases}$$

Translations

 * French:
 * German: Primorial, Primfakultät
 * Italian:
 * Turkish: primoriyel