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 former interpretation.

The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the 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 five primorials $p n #$ are:

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 $o ( )$ is 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 $π (n)$ 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=0,\ 1\\(n-1)\#\times n&{\text{if }}n{\text{ is prime}}\\(n-1)\#&{\text{if }}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 $π (12) = 5$ , this can be calculated as:

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

Consider the first 12 values of $n #$ :

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 $12# = p 5 # = 11#$ since 12 is a composite number.

Primorials are related to the first Chebyshev function, written $ϑ (n)$ or $θ (n)$ according to:

\ln(n\#)=\vartheta (n).

^[4]

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

n\#=e^{(1+o(1))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, 7009223613394100000♠2236133941 + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 7009513634125100000♠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 $φ (n) / n$ is smaller than it for any lesser integer, 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.^[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 function and Jordan's totient function $J k (n)$ :

\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

$n$	$n #$	$p n$	$p n #$ ^[10]	Primorial prime?
$n$	$n #$	$p n$	$p n #$ ^[10]	p_n# + 1^[11]	p_n# − 1^[12]
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	7003231000000000000♠2310	Yes	Yes
6	30	13	7004300300000000000♠30030	No	Yes
7	210	17	7005510510000000000♠510510	No	No
8	210	19	7006969969000000000♠9699690	No	No
9	210	23	7008223092870000000♠223092870	No	No
10	210	29	7009646969323000000♠6469693230	No	No
11	7003231000000000000♠2310	31	7011200560490130000♠200560490130	Yes	No
12	7003231000000000000♠2310	37	7012742073813481000♠7420738134810	No	No
13	7004300300000000000♠30030	41	7014304250263527210♠304250263527210	No	Yes
14	7004300300000000000♠30030	43	7016130827613316700♠13082761331670030	No	No
15	7004300300000000000♠30030	47	7017614889782588491♠614889782588491410	No	No
16	7004300300000000000♠30030	53	7019325891584771900♠32589158477190044730	No	No
17	7005510510000000000♠510510	59	7021192276035015421♠1922760350154212639070	No	No
18	7005510510000000000♠510510	61	7023117288381359406♠117288381359406970983270	No	No
19	7006969969000000000♠9699690	67	7024785832155108026♠7858321551080267055879090	No	No
20	7006969969000000000♠9699690	71	7026557940830126698♠557940830126698960967415390	No	No
21	7006969969000000000♠9699690	73	7028407296805992490♠40729680599249024150621323470	No	No
22	7006969969000000000♠9699690	79	7030321764476734067♠3217644767340672907899084554130	No	No
23	7008223092870000000♠223092870	83	7032267064515689275♠267064515689275851355624017992790	No	No
24	7008223092870000000♠223092870	89	7034237687418963455♠23768741896345550770650537601358310	No	No
25	7008223092870000000♠223092870	97	7036230556796394551♠2305567963945518424753102147331756070	No	No
26	7008223092870000000♠223092870	101	7038232862364358497♠232862364358497360900063316880507363070	No	No
27	7008223092870000000♠223092870	103	7040239848235289252♠23984823528925228172706521638692258396210	No	No
28	7008223092870000000♠223092870	107	7042256637611759499♠2566376117594999414479597815340071648394470	No	No
29	7009646969323000000♠6469693230	109	7044279734996817854♠279734996817854936178276161872067809674997230	No	No
30	7009646969323000000♠6469693230	113	7046316100546404176♠31610054640417607788145206291543662493274686990	No	No
31	7011200560490130000♠200560490130	127	7048401447693933303♠4014476939333036189094441199026045136645885247730	No	No
32	7011200560490130000♠200560490130	131	7050525896479052627♠525896479052627740771371797072411912900610967452630	No	No
33	7011200560490130000♠200560490130	137	7052720478176302100♠72047817630210000485677936198920432067383702541010310	No	No
34	7011200560490130000♠200560490130	139	7055100146466505991♠10014646650599190067509233131649940057366334653200433090	No	No
35	7011200560490130000♠200560490130	149	7057149218235093927♠1492182350939279320058875736615841068547583863326864530410	No	No
36	7011200560490130000♠200560490130	151	7059225319534991831♠225319534991831177328890236228992001350685163362356544091910	No	No
37	7012742073813481000♠7420738134810	157	7061353751669937174♠35375166993717494840635767087951744212057570647889977422429870	No	No
38	7012742073813481000♠7420738134810	163	7063576615221997595♠5766152219975951659023630035336134306565384015606066319856068810	No	No
39	7012742073813481000♠7420738134810	167	7065962947420735983♠962947420735983927056946215901134429196419130606213075415963491270	No	No
40	7012742073813481000♠7420738134810	173	7068166589903787325♠166589903787325219380851695350896256250980509594874862046961683989710	No	No

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, N.J.A. (ed.). "Sequence 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 0819198. 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, N.J.A. (ed.). "Sequence 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.
↑ http://planetmath.org/TableOfTheFirst100Primorials
↑ Sloane, N.J.A. (ed.). "Sequence A014545 (Primorial plus 1 prime indices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
↑ Sloane, N.J.A. (ed.). "Sequence A057704 (Primorial - 1 prime indices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

References

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

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[mathworld-1] 1 2 Weisstein, Eric W. "Primorial". MathWorld.

[OEIS_A002110-2] 1 2 (sequence A002110 in the OEIS)

[OEIS_A034386-3] (sequence A034386 in the OEIS)

[4] Weisstein, Eric W. "Chebyshev Functions". MathWorld.

[5] Sloane, N.J.A. (ed.). "Sequence 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 0819198. Zbl 0538.10006.

[Wells_2011-7] Wells, David (2011). Prime Numbers: The Most Mysterious Figures in Math. John Wiley & Sons. p. 29. ISBN 9781118045718. Retrieved 16 March 2016.

[8] Sloane, N.J.A. (ed.). "Sequence A036691 (Compositorial numbers: product of first n composite numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[mezo-9] Mező, István (2013). "The Primorial and the Riemann zeta function". The American Mathematical Monthly. 120 (4): 321.

[10] ttp://planetmath.org/TableOfTheFirst100Primorials

[11] Sloane, N.J.A. (ed.). "Sequence A014545 (Primorial plus 1 prime indices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[12] Sloane, N.J.A. (ed.). "Sequence A057704 (Primorial - 1 prime indices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.