# Primorial

Not to be confused with primordial (disambiguation).
pn# as a function of n, plotted logarithmically.
n# as a function of n (red dots), compared to n!. Both plots are logarithmic.

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 successively multiplying positive integers, only 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 as at its predecessor). The rest of this article uses the latter interpretation.

The name "primorial", attributed to Harvey Dubner, draws an analogy to primes the same way the name "factorial" relates to factors.

## Definition for prime numbers

For the nth prime number pn the primorial pn# is defined as the product of the first n primes:[1][2]

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

where pk 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 six primorials pn# are:

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

The sequence also includes p0# = 1 as empty product.

Asymptotically, primorials pn# grow according to:

$p_n\# = e^{(1 + o(1)) n \log n},$

where $o(\cdot)$ is the little-o notation.[2]

## Definition for natural numbers

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]

$n\# = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\#$

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

This is equivalent to:

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

For example, 12# represents the product of those primes ≤ 12:

$12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.$

Since $\scriptstyle\pi(12)=5$, this can be calculated as:

$12\# = p_{\pi(12)}\# = p_5\# = 2310.$

Consider the first 12 primorials n#:

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

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

The natural logarithm of n# is the first Chebyshev function, written $\theta(n)$ or $\thetasym(n)$, which approaches the linear n for large n.[4]

Primorials n# grow according to:

$\ln (n\#) \sim n.$

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, 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).[5]

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.

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.[6]

## Appearance

The Riemann zeta function at positive integers greater than one can be expressed[7] by using the primorial and the $J_k(n)$ Jordan's totient function:

$\zeta(k)=\frac{2^k}{2^k-1}+\sum_{r=2}^\infty\frac{(p_{r-1}\#)^k}{J_k(p_r\#)},\quad k=2,3,\dots$

## Table of primorials

n n# pn pn#
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
16 30030 53 32589158477190044730
17 510510 59 1922760350154212639070
18 510510 61 117288381359406970983270
19 9699690 67 7858321551080267055879090
20 9699690 71 557940830126698960967415390

## Notes

1. ^ a b
2. ^ a b (sequence A002110 in OEIS)
3. ^ (sequence A034386 in OEIS)
4. ^
5. ^ "Sloane's A002182 : Highly composite numbers", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. ^ Masser, D.W.; Shiu, P. (1986). "On sparsely totient numbers". Pac. J. Math. 121: 407–426. ISSN 0030-8730. MR 819198. Zbl 0538.10006.
7. ^ Mező, István (2013). "The Primorial and the Riemann zeta function". The American Mathematical Monthly 120 (4): 321.

## References

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