page_title: Primorial
section_title: Definition for natural numbers
hierarchical_section_title: Primorial / Definition for natural numbers
context_page_description: In mathematics, and more particularly in number theory, primorial, denoted by “#”, is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.
  The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors.
context_section_description: 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:
  For example, 12# represents the product of those primes ≤ 12:
  Since π(12) = 5, this can be calculated as:
  Consider the first 12 values of n#:
  1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.
  We see that for composite n every term n# simply duplicates the preceding term (n − 1)#, as given in the definition. In the above example we have 12# = p₅# = 11# since 12 is a composite number.
  Primorials are related to the first Chebyshev function, written ϑ(n) or θ(n) according to:
  Since ϑ(n) asymptotically approaches n for large values of n, primorials therefore grow according to:
  The idea of multiplying all known primes occurs in some proofs of the infinitude of the prime numbers, where it is used to derive the existence of another prime.
caption: n# as a function of n (red dots), compared to n!, both plotted logarithmically.
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