Primorial

Not to be confused with primordial (disambiguation).

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", coined by Harvey Dubner, draws an analogy to primes the same way the name "factorial" relates to factors.

Definition for primorial numbers

For the $n$ th prime number $p_{n}$ the primorial $p_{n}\#$ is defined as the product of the first $n$ primes:^[1]^[2]

p_{n}\#\equiv \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 \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 the OEIS).

The sequence also includes $p_{0}\#=1$ 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 $n$ , a primorial $n\#$ can also be defined, namely as the product of those primes ≤ $n$ :^[1]^[3]

n\#\equiv \prod _{i=1}^{\pi (n)}p_{i}=p_{\pi (n)}\#

where $\pi (n)$ is the prime-counting function (sequence A000720 in the 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 $\pi (12)=5$ , 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 $12\#=p_{5}\#=11\#$ 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

$n$	$n\#$	$p_{n}$	$p_{n}\#$
0	1	no prime	1
1	1	2	2
2	2	3	6
3	6	5	30
4	6	7	210
5	30	11	2 310
6	30	13	30 030
7	210	17	510 510
8	210	19	9 699 690
9	210	23	223 092 870
10	210	29	6 469 693 230
11	2 310	31	200 560 490 130
12	2 310	37	7 420 738 134 810
13	30 030	41	304 250 263 527 210
14	30 030	43	13 082 761 331 670 030
15	30 030	47	614 889 782 588 491 410
16	30 030	53	32 589 158 477 190 044 730
17	510 510	59	1 922 760 350 154 212 639 070
18	510 510	61	117 288 381 359 406 970 983 270
19	9 699 690	67	7 858 321 551 080 267 055 879 090
20	9 699 690	71	557 940 830 126 698 960 967 415 390

Notes

1 2 Weisstein, Eric W. "Primorial". MathWorld.
1 2 (sequence A002110 in the OEIS)
↑ (sequence A034386 in the OEIS)
↑ Weisstein, Eric W. "Chebyshev Functions". MathWorld.
↑ "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
↑ 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.
↑ Wells, David (2011). Prime Numbers: The Most Mysterious Figures in Math. John Wiley & Sons. p. 29. ISBN 9781118045718. Retrieved 16 March 2016.
↑ "Sloane's A036691 : Compositorial numbers: product of first n composite numbers.". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
↑ 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.

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