Primorial

p_n# 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 Primorial numbers

For the nth prime number p_n 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 six primorials p_n# are:

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

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

Asymptotically, primorials p_n# 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]

The n-compositorial of a composite number n is the product of all composite numbers up to and including n.^[7] The n-compositorial is equal to the n-factorial divided by the primorial n#. The compositorials are 1, 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, ...^[8]

Appearance

The Riemann zeta function at positive integers greater than one can be expressed^[9] 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

See also

• Bonse's inequality
• Factorial number system
• Primorial prime

Notes

1. Weisstein, Eric W., "Primorial", MathWorld.
2. (sequence A002110 in OEIS)
3. (sequence A034386 in OEIS)
4. Weisstein, Eric W., "Chebyshev Functions", MathWorld.
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. doi:10.2140/pjm.1986.121.407. ISSN 0030-8730. MR 819198. Zbl 0538.10006.
7. Wells, David (2011). Prime Numbers: The Most Mysterious Figures in Math. John Wiley & Sons. p. 29. ISBN 9781118045718. Retrieved 16 March 2016.
8. "Sloane's A036691 : Compositorial numbers: product of first n composite numbers.", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
9. Mező, István (2013). "The Primorial and the Riemann zeta function". The American Mathematical Monthly 120 (4): 321.

References

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