Primorial

In mathematics, and more particularly in number theory, primorial, denoted by "pn#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.

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

## Contents

* 1 Definition for prime numbers
* 2 Definition for natural numbers
* 3 Properties
* 4 Applications
* 5 Compositorial
* 6 Riemann zeta function
* 7 Table of primorials
* 8 See also
* 9 Notes
* 10 References

## Definition for prime numbers

pn# as a function of n, plotted logarithmically.

The primorial pn# is defined as the product of the first n primes:[1][2]

pn#=∏k=1npk,

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

p5#=2×3×5×7×11=2310.

The first few primorials pn# are:

1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690... (sequence A002110 in the OEIS).

Asymptotically, primorials grow according to[2]

pn#=e(1+o(1))nlog⁡n.

## Definition for natural numbers

n! (yellow) as a function of n, compared to n# (red), both plotted logarithmically.

In general, for a positive integer n, its primorial n# is the product of all primes less than or equal to n; that is,[1][3]

n#=∏p≤npprimep=∏i=1π(n)pi=pπ(n)#,

where π(n) is the prime-counting function (sequence A000720 in the OEIS). This is equivalent to

n#={1if n=0, 1(n−1)#×nif n is prime(n−1)#if n is composite.

For example, 12# represents the product of all primes no greater than 12:

12#=2×3×5×7×11=2310.

Since π(12)=5, this can be calculated as:

12#=pπ(12)#=p5#=2310.

Consider the first 12 values of the sequence n#:

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

We see that for composite n, every term n# is equal to the preceding term (n−1)#. In the above example we have 12#=p5#=11# since 12 is composite.

Primorials are related to the first Chebyshev function ϑ(n) by[4]

ln⁡(n#)=ϑ(n).

Since ϑ(n) asymptotically approaches n for large values of n, primorials therefore grow according to:

n#=e(1+o(1))n.

## Properties

* For any n,p∈ℕ, n#=p# iff p is the largest prime such that p≤n.
* Let pk be the kth prime. Then pk# has exactly 2k divisors.
* The sum of the reciprocal values of the primorial converges towards a constant

∑pprime1p#=12+16+130+…=0.7052301717918… (sequence A064648 in the OEIS)

The Engel expansion of this number results in the sequence of the prime numbers. Griffiths (2015) proved that it is irrational.[5]

* Euclid's proof of his theorem on the infinitude of primes can be paraphrased by saying that, for any prime p, the number p#+1 has a prime divisor not contained in the set of primes less than or equal to p.
* limn->∞n#n=e. For n<1011, the values are smaller than e,[6] but for larger n, the values of the function exceed e and oscillate infinitely around e later on.
* Since the binomial coefficient (2nn) is divisible by every prime between n+1 and 2n, and since (2nn)≤4n, we have the following upper bound:[7] n#≤4n.
* Using elementary methods, Denis Hanson showed that n#≤3n.[8]
* Using more advanced methods, Rosser and Schoenfeld showed that n#≤(2.763)n.[9] Furthermore, they showed that for n≥563, n#≥(2.22)n.[9]

## Applications

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.[10]

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 positive integer less than n, 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.

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.[11]

## Compositorial

The n-compositorial of a composite number n is the product of all composite numbers up to and including n.[12] 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, ...[13]

## Riemann zeta function

The Riemann zeta function at positive integers greater than one can be expressed[14] by using the primorial function and Jordan's totient function Jk:

ζ(k)=2k2k−1+∑r=2∞(pr−1#)kJk(pr#),k∈ℤ>1.

## Table of primorials

n | n# | pn | pn# | Primorial prime?

|
|
|
|

pn# + 1[15] | pn# − 1[16]
0  | 1  | N/A  | 1 | Yes  | No
1  | 1  | 2  | 2  | Yes  | No
2  | 2  | 3  | 6  | Yes  | Yes
3  | 6  | 5  | 30  | Yes  | Yes
4  | 6  | 7  | 210  | Yes  | No
5  | 30  | 11  | 2310 | Yes  | Yes
6  | 30  | 13  | 30030 | No  | Yes
7  | 210  | 17  | 510510 | No  | No
8  | 210  | 19  | 9699690 | No  | No
9  | 210  | 23  | 223092870 | No  | No
10  | 210  | 29  | 6469693230 | No  | No
11  | 2310 | 31  | 200560490130 | Yes  | No
12  | 2310 | 37  | 7420738134810 | No  | No
13  | 30030 | 41  | 304250263527210 | No  | Yes
14  | 30030 | 43  | 13082761331670030 | No  | No
15  | 30030 | 47  | 614889782588491410 | No  | No
16  | 30030 | 53  | 32589158477190044730 | No  | No
17  | 510510 | 59  | 1922760350154212639070 | No  | No
18  | 510510 | 61  | 117288381359406970983270 | No  | No
19  | 9699690 | 67  | 7858321551080267055879090 | No  | No
20  | 9699690 | 71  | 557940830126698960967415390 | No  | No
21  | 9699690 | 73  | 40729680599249024150621323470 | No  | No
22  | 9699690 | 79  | 3217644767340672907899084554130 | No  | No
23  | 223092870 | 83  | 267064515689275851355624017992790 | No  | No
24  | 223092870 | 89  | 23768741896345550770650537601358310 | No  | Yes
25  | 223092870 | 97  | 2305567963945518424753102147331756070 | No  | No
26  | 223092870 | 101  | 232862364358497360900063316880507363070 | No  | No
27  | 223092870 | 103  | 23984823528925228172706521638692258396210 | No  | No
28  | 223092870 | 107  | 2566376117594999414479597815340071648394470 | No  | No
29  | 6469693230 | 109  | 279734996817854936178276161872067809674997230 | No  | No
30  | 6469693230 | 113  | 31610054640417607788145206291543662493274686990 | No  | No
31  | 200560490130 | 127  | 4014476939333036189094441199026045136645885247730 | No  | No
32  | 200560490130 | 131  | 525896479052627740771371797072411912900610967452630 | No  | No
33  | 200560490130 | 137  | 72047817630210000485677936198920432067383702541010310 | No  | No
34  | 200560490130 | 139  | 10014646650599190067509233131649940057366334653200433090 | No  | No
35  | 200560490130 | 149  | 1492182350939279320058875736615841068547583863326864530410 | No  | No
36  | 200560490130 | 151  | 225319534991831177328890236228992001350685163362356544091910 | No  | No
37  | 7420738134810 | 157  | 35375166993717494840635767087951744212057570647889977422429870 | No  | No
38  | 7420738134810 | 163  | 5766152219975951659023630035336134306565384015606066319856068810 | No  | No
39  | 7420738134810 | 167  | 962947420735983927056946215901134429196419130606213075415963491270 | No  | No
40  | 7420738134810 | 173  | 166589903787325219380851695350896256250980509594874862046961683989710 | No  | No

## See also

* Bonse's inequality
* Chebyshev function
* Primorial number system
* Primorial prime

## Notes

1. ↑ 1.0 1.1 Weisstein, Eric W.. "Primorial". http://mathworld.wolfram.com/Primorial.html.
2. ↑ 2.0 2.1 (sequence A002110 in the OEIS)
3. ↑ (sequence A034386 in the OEIS)
4. ↑ Weisstein, Eric W.. "Chebyshev Functions". http://mathworld.wolfram.com/ChebyshevFunctions.html.
5. ↑ Griffiths, Martin (November 2015). "99.29 On the sum of the reciprocals of the primorials". The Mathematical Gazette 99 (546): 522–523. doi:10.1017/mag.2015.91.
6. ↑ L. Schoenfeld: Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II. Math. Comp. Vol. 34, No. 134 (1976) 337–360; p. 359.
Cited in: G. Robin: Estimation de la fonction de Tchebychef θ sur le k-ieme nombre premier et grandes valeurs de la fonction ω(n), nombre de diviseurs premiers de n. Acta Arithm. XLII (1983) 367–389 (PDF 731KB); p. 371
7. ↑ G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. 4th Edition. Oxford University Press, Oxford 1975. ISBN 0-19-853310-1.
Theorem 415, p. 341
8. ↑ Hanson, Denis (March 1972). "On the Product of the Primes". Canadian Mathematical Bulletin 15 (1): 33–37. doi:10.4153/cmb-1972-007-7. ISSN 0008-4395.
9. ↑ 9.0 9.1 Rosser, J. Barkley; Schoenfeld, Lowell (1962-03-01). "Approximate formulas for some functions of prime numbers". Illinois Journal of Mathematics 6 (1). doi:10.1215/ijm/1255631807. ISSN 0019-2082.
10. ↑ Sloane, N. J. A., ed. "Sequence A002182 (Highly composite numbers)". OEIS Foundation. https://oeis.org/A002182.
11. ↑ Masser, D.W.; Shiu, P. (1986). "On sparsely totient numbers". Pacific Journal of Mathematics 121 (2): 407–426. doi:10.2140/pjm.1986.121.407. ISSN 0030-8730. http://projecteuclid.org/euclid.pjm/1102702441.
12. ↑ Wells, David (2011). Prime Numbers: The Most Mysterious Figures in Math. John Wiley & Sons. p. 29. ISBN 9781118045718. https://books.google.com/books?id=1MTcYrbTdsUC&q=Compositorial+primorial&pg=PA29. Retrieved 16 March 2016.
13. ↑ Sloane, N. J. A., ed. "Sequence A036691 (Compositorial numbers: product of first n composite numbers.)". OEIS Foundation. https://oeis.org/A036691.
14. ↑ Mező, István (2013). "The Primorial and the Riemann zeta function". The American Mathematical Monthly 120 (4): 321.
15. ↑ Sloane, N. J. A., ed. "Sequence A014545 (Primorial plus 1 prime indices)". OEIS Foundation. https://oeis.org/A014545.
16. ↑ Sloane, N. J. A., ed. "Sequence A057704 (Primorial - 1 prime indices)". OEIS Foundation. https://oeis.org/A057704.

## References

* Dubner, Harvey (1987). "Factorial and primorial primes". J. Recr. Math. 19: 197–203.
* Spencer, Adam "Top 100" Number 59 part 4.

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