Primorial

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p_n# as a function of n, plotted logarithmically.

n# as a function of n (red dots), compared to n!. Both plots are logarithmic.

For n ≥ 1, 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:^[1]^[2]

$p_n\# = \prod_{k=1}^n p_k$

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

$p_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310$

The first few primorials p_n# are:

1, 2, 6, 30, 210, 2310 (sequence A002110 in OEIS)

Asymptotically, primorials p_n# grow according to:

$p_n\# = \exp\left((1 + o(1)) \cdot n \log n\right),$

where "exp" is the exponential function e^x and "o" is the "little-o" notation (see Big O notation).^[2] Its natural logarithm is the first Chebyshev function, written $θ(n)$ or $\thetasym(n)$ , which approaches the linear n for large n.^[3]

In contrast, n# is defined as the product of those primes ≤ n:^[1]^[4]

$n\# = \begin{cases} 1 & n = 1 \\ n \times ((n-1)\#) & n > 1 \And n \text{ prime} \\ (n-1)\# & n > 1 \And n \text{ composite} \end{cases}$

This is equivalent to:^[4]

$n\# = p_{\pi(n)}\#$

where, π(n) is the prime-counting function (sequence A000720 in OEIS), giving the number of primes ≤ n.

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

$7\# = 2 \times 3 \times 5 \times 7$

As noted, this can be calculated as:

$7\# = p_{\pi(7)}\,\!$

$\pi(7) = 4\,\!$

$7\# = p_4\# = 210\,\!$

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 $φ(n) / n$ 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.

[edit] Table of primorials

n	p_n#	n#
1	1	1
2	2	2
3	6	6
4	30	6
5	210	30
6	2310	30
7	30030	210
8	510510	210
9	9699690	210
10	223092870	210
11	6469693230	2310
12	200560490130	2310
13	7420738134810	30030
14	304250263527210	30030
15	13082761331670030	30030