Primorial

The primorial has two similar but distinct meanings. The name is attributed to Harvey Dubner and is a portmanteau of "prime" and "factorial". The primorial "p_n#" is defined as the product of the first "n" primes:Mathworld | urlname=Primorial | title=Primorial] OEIS|id=A002110]

:$p\_n\#\; =\; prod\_\{k=1\}^n\; p\_k$

where "p_k" is the "k"th prime number. For instance, "p"₅# signifies the product of the first 5 primes:

:$p\_5\#\; =\; 2\; imes\; 3\; imes\; 5\; imes\; 7\; imes\; 11\; =\; 2310.$

The first few primorials "p_n#" are:

:1, 2, 6, 30, 210, 2310. OEIS|id=A002110

The sequence also includes "p"₀# = 1 as empty product.

Asymptotically, primorials "p_n#" grow according to:

:$p\_n\#\; =\; exp\; left\; [\; (1\; +\; o(1))\; cdot\; n\; log\; n\; ight\; ]\; ,$

where "exp" is the exponential function "e"^"x" and "o" is the "little-o" notation (see Big O notation). Its natural logarithm is the first Chebyshev function, written $heta(n)$ or $hetasym(n)$, which approaches the linear "n" for large "n". [Mathworld | urlname=ChebyshevFunctions | title=Chebyshev Functions]

In contrast, "n#" is defined as the product of those primes ≤ "n", for "n" ≥ 1:OEIS|id=A034386]

:

This is equivalent to:

:$n\#\; =\; p\_\{pi(n)\}\#$

where, π(n) is the prime-counting function OEIS|id=A000720, giving the number of primes ≤ "n".

For example, 7# represents the product of those primes ≤ 7:

:$7\#\; =\; 2\; imes\; 3\; imes\; 5\; imes\; 7\; =\; 210.$

Since π(7) = 4, this can be calculated as:

:$7\#\; =\; p\_\{pi(7)\}\#\; =\; p\_4\#\; =\; 210.$

The first primorials "n#" are:

:1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310.

Note that every term n# for composite "n" simply duplicates the preceding term ("n"−1)#, as evident in the definition given.

Primorials "n#" grow according to::$log\; n\#\; sim\; n.$

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 square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial "n", the fraction $phi(n)/n$ is smaller than for any lesser integer, where $phi$ 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.

Table of primorials

See also

* Primorial prime

Notes

References

* Harvey Dubner, "Factorial and primorial primes". " J. Recr. Math.", 19, 197–203, 1987.