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 multiplying successive positive integers, only successive 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 for its predecessor). The name "primorial" is attributed to Harvey Dubner, and is a portmanteau of prime and factorial.
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For the nth prime number p_{n} the primorial p_{n}# is defined as the product of the first n primes:^{[1]}^{[2]}
where p_{k} is the kth prime number.
For instance, p_{5}# signifies the product of the first 5 primes:
The first few primorials p_{n}# are:
The sequence also includes p_{0}# = 1 as empty product.
Asymptotically, primorials p_{n}# grow according to:
where "exp" is the exponential function e^{x} and "o" is the littleo notation.^{[2]}
In general, for a positive integer n such a primorial n# can also be defined, namely as the product of those primes ≤ n:^{[1]}^{[3]}
where, π(n) is the primecounting function (sequence A000720 in 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 primorials n#:
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 that 12# = p_{5}# = 11#, since 12 is a composite number.
Natural logarithm of n# is the first Chebyshev function, written θ(n) or , which approaches the linear n for large n.^{[4]}
Primorials n# grow according to:
The idea of multiplying all known primes occurs in a proof of the infinitude of the prime numbers; it is applied to show a contradiction in the idea that the primes could be finite in number.
Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, 2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 5136341251. 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes.
Every highly composite number is a product of primorials (e.g. 360 = 2·6·30).
Primorials are all squarefree 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.
Any completely multiplicative function is defined by its values at primorials, since it is defined by its values at primes, which can be recovered by division of adjacent values.
n  n#  p_{n}  p_{n}# 

0  1  no prime  1 
1  1  2  2 
2  2  3  6 
3  6  5  30 
4  6  7  210 
5  30  11  2310 
6  30  13  30030 
7  210  17  510510 
8  210  19  9699690 
9  210  23  223092870 
10  210  29  6469693230 
11  2310  31  200560490130 
12  2310  37  7420738134810 
13  30030  41  304250263527210 
14  30030  43  13082761331670030 
15  30030  47  614889782588491410 
