Primorial

Definition for primorial numbers

For the nth prime number p_n, the primorial p_n# is defined as the product of the first n primes:

where p_k is the kth prime number. For instance, p₅# signifies the product of the first 5 primes:

The first six primorials p_n# are:

The sequence also includes p₀# = 1 as empty product. Asymptotically, primorials p_n# grow according to:

where o( ) is little-o notation.

Definition for natural numbers

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 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 12# = p₅# = 11# since 12 is a composite number.

The natural logarithm of n# is the first Chebyshev function, written ϑ(n) or θ(n), which approaches the linear n for large n.

Primorials n# 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.

Applications and properties

Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, 7009223613394100000♠2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 7009513634125100000♠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 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 it 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.

Base systems corresponding to primorials (such as base 30, not to be confused with the primorial number system) have a lower proportion of repeating fractions than any smaller base.

Every primorial is a sparsely totient number.

The n-compositorial of a composite number n is the product of all composite numbers up to and including n. The n-compositorial is equal to the n-factorial divided by the primorial n#. The compositorials are

Appearance

The Riemann zeta function at positive integers greater than one can be expressed by using the primorial function and Jordan's totient function J_k(n):