Primorial

In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, only prime numbers are multiplied.

There are two conflicting definitions that differ in the interpretation of the argument: the first interprets the argument as an index into the sequence of prime numbers (so that the function is strictly increasing), while the second interprets the argument as a bound on the prime numbers to be multiplied (so that the function value at any composite number is the same as at its predecessor). The rest of this article uses the latter interpretation.

The name "primorial", coined by Harvey Dubner, draws an analogy to primes the same way the name "factorial" relates to factors.

For the $n$ th prime number $p n$ , the primorial $p n #$ is defined as the product of the first $n$ primes:

where $p k$ is the $k$ th prime number. For instance, $p 5 #$ signifies the product of the first 5 primes:

The first six primorials $p n #$ are:

The sequence also includes $p 0 # = 1$ as empty product. Asymptotically, primorials $p n #$ grow according to:

where $o ( )$ is little- $o$ notation.

In general for a positive integer $n$ , a primorial $n #$ can also be defined, namely as the product of those primes ≤ $n$ :

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