Primorial

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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.

Contents

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

Definition for prime numbers

pn# as a function of n, plotted logarithmically. Primorial pn plot.png
pn# as a function of n, plotted logarithmically.

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

where is the -th prime number. For instance, signifies the product of the first 5 primes:

The first few primorials are:

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

Asymptotically, primorials grow according to [2]

Comparison

The table below shows the comparison between and .

Absolute errorRelative error
11101
22422
3627214.5
4302562268.53...

As can be seen, the absolute error and relative error diverges to infinity.

Definition for natural numbers

n
!
{\displaystyle n!}
(yellow) as a function of
n
{\displaystyle n}
, compared to
n
#
{\displaystyle n\#}
(red), both plotted logarithmically. Primorial n plot.png
(yellow) as a function of , compared to (red), both plotted logarithmically.

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

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

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

Since , this can be calculated as:

Consider the first 12 values of the sequence :

We see that for composite , every term is equal to the preceding term . In the above example we have since 12 is composite.

Primorials are related to the first Chebyshev function by [4]

Since asymptotically approaches for large values of , primorials therefore grow according to:

Properties

The Engel expansion of this number results in the sequence of the prime numbers (sequence A064648 in the OEIS ).

Applications

Primorials play a role in the search for prime numbers in additive arithmetic progressions. For instance, results in a prime, beginning a sequence of thirteen primes found by repeatedly adding , and ending with . is also the common difference in arithmetic progressions of fifteen and sixteen primes.

Every highly composite number is a product of primorials. [9]

Primorials are all square-free integers, and each one has more distinct prime factors than any number smaller than it. For each primorial , the fraction is smaller than for any lesser integer,[ clarification needed ] where 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. [10]

Compositorial

The n-compositorial of a composite number n is the product of all composite numbers up to and including n. [11] 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, ... [12]

Riemann zeta function

The Riemann zeta function at positive integers greater than one can be expressed [13] by using the primorial function and Jordan's totient function :

.

Table of primorials

nn#pnpn# Primorial prime?
pn# + 1 [14] pn# − 1 [15]
01 1 YesNo
1122YesNo
2236YesYes
36530YesYes
467210YesNo
530112310YesYes
6301330030NoYes
721017510510NoNo
8210199699690NoNo
921023223092870NoNo
10210296469693230NoNo
11231031200560490130YesNo
122310377420738134810NoNo
133003041304250263527210NoYes
14300304313082761331670030NoNo
153003047614889782588491410NoNo
16300305332589158477190044730NoNo
17510510591922760350154212639070NoNo
1851051061117288381359406970983270NoNo
199699690677858321551080267055879090NoNo
20969969071557940830126698960967415390NoNo
2196996907340729680599249024150621323470NoNo
229699690793217644767340672907899084554130NoNo
2322309287083267064515689275851355624017992790NoNo
242230928708923768741896345550770650537601358310NoYes
25223092870972305567963945518424753102147331756070NoNo
26223092870101232862364358497360900063316880507363070NoNo
2722309287010323984823528925228172706521638692258396210NoNo
282230928701072566376117594999414479597815340071648394470NoNo
296469693230109279734996817854936178276161872067809674997230NoNo
30646969323011331610054640417607788145206291543662493274686990NoNo
312005604901301274014476939333036189094441199026045136645885247730NoNo
32200560490130131525896479052627740771371797072411912900610967452630NoNo
3320056049013013772047817630210000485677936198920432067383702541010310NoNo
3420056049013013910014646650599190067509233131649940057366334653200433090NoNo
352005604901301491492182350939279320058875736615841068547583863326864530410NoNo
36200560490130151225319534991831177328890236228992001350685163362356544091910NoNo
37742073813481015735375166993717494840635767087951744212057570647889977422429870NoNo
3874207381348101635766152219975951659023630035336134306565384015606066319856068810NoNo
397420738134810167962947420735983927056946215901134429196419130606213075415963491270NoNo
407420738134810173166589903787325219380851695350896256250980509594874862046961683989710NoNo

See also

Notes

  1. 1 2 Weisstein, Eric W. "Primorial". MathWorld .
  2. 1 2 (sequence A002110 in the OEIS )
  3. (sequence A034386 in the OEIS )
  4. Weisstein, Eric W. "Chebyshev Functions". MathWorld .
  5. L. Schoenfeld: Sharper bounds for the Chebyshev functions and . II. Math. Comp. Vol. 34, No. 134 (1976) 337–360; p. 359.
    Cited in: G. Robin: Estimation de la fonction de Tchebychef sur le k-ieme nombre premier et grandes valeurs de la fonction , nombre de diviseurs premiers de n. Acta Arithm. XLII (1983) 367–389 (PDF 731KB); p. 371
  6. 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
  7. 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.
  8. 1 2 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.
  9. Sloane, N. J. A. (ed.). "SequenceA002182(Highly composite numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  10. 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. MR   0819198. Zbl   0538.10006.
  11. Wells, David (2011). Prime Numbers: The Most Mysterious Figures in Math. John Wiley & Sons. p. 29. ISBN   9781118045718 . Retrieved 16 March 2016.
  12. Sloane, N. J. A. (ed.). "SequenceA036691(Compositorial numbers: product of first n composite numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  13. Mező, István (2013). "The Primorial and the Riemann zeta function". The American Mathematical Monthly. 120 (4): 321.
  14. Sloane, N. J. A. (ed.). "SequenceA014545(Primorial plus 1 prime indices)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  15. Sloane, N. J. A. (ed.). "SequenceA057704(Primorial - 1 prime indices)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.

References