Primorial

The primorial, a portmanteau of prime and factorial, is formally defined as

$$p_n \# = \prod^{n}_{i = 1} p_i$$

where $$p_n$$ is the nth prime.

Another slightly more complex definition, which expands the domain of the function beyond prime numbers, is

$$n \# = \prod^{\pi (n)}_{i = 1} p_i$$

where $$p_n$$ is the nth prime and $$\pi (n)$$ is the prime counting function.

Using either definition, the primorial of n can be informally defined as "the product of all prime numbers up to n, inclusive." For example, $$16 \# = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 = 30030$$.

The primorial's relationship to the Chebyshev function $$\theta (x)$$ gives it the property

$$\lim_{n\rightarrow\infty} \sqrt[p_n]{p_n \#} = e$$

where e is the mathematical constant.