Primorial

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

[2] In general, for a positive integer n, its primorial, n#, is the product of the primes that are not greater than n; that is,[1][3] where π(n) is the prime-counting function (sequence A000720 in the OEIS), which gives 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#: We see that for composite n every term n# simply duplicates the preceding term (n − 1)#, as given in the definition.

Notes: Primorials play a role in the search for prime numbers in additive arithmetic progressions.

[9] Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it.

For each primorial n, the fraction ⁠φ(n)/n⁠ is smaller than for any lesser integer, where φ is the Euler totient function.