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. The symbol for primorial is “#”. 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. The symbol for primorial is “#”. The name 'primorial', coined by Harvey Dubner, draws an analogy to primes similar to the way the name 'factorial' relates to factors. For the nth prime number pn, the primorial pn# is defined as the product of the first n primes: where pk is the kth prime number. For instance, p5# signifies the product of the first 5 primes: The first five primorials pn# are: The sequence also includes p0# = 1 as empty product. Asymptotically, primorials pn# 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: where π(n) is the prime-counting function (sequence A000720 in the OEIS), giving the number of primes ≤ n. This is equivalent to: