In mathematics, and more particularly in number theory, primorial, denoted by "#", 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.
For the nth prime number pn, the primorial pn# is defined as the product of the first n primes: [1] [2]
where pk is the kth prime number. For instance, p5# signifies the product of the first 5 primes:
The first five primorials pn# are:
The sequence also includes p0# = 1 as empty product. Asymptotically, primorials pn# grow according to:
where o( ) is Little O notation. [2]
In general, for a positive integer n, its primorial, n#, is the product of the primes that are not greater than n; that is, [1] [3]
where π(n) is the prime-counting function (sequence A000720 in the OEIS ), which gives the number of primes ≤ n. This is equivalent to:
For example, 12# represents the product of those primes ≤ 12:
Since π(12) = 5, this can be calculated as:
Consider the first 12 values of n#:
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# = p5# = 11# since 12 is a composite number.
Primorials are related to the first Chebyshev function, written ϑ(n) or θ(n) according to:
Since ϑ(n) asymptotically approaches n for large values of n, primorials therefore grow according to:
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.
Notes:
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). [9]
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 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. [10]
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
The Riemann zeta function at positive integers greater than one can be expressed [13] by using the primorial function and Jordan's totient function Jk(n):
n | n# | pn | pn# | Primorial prime? | |
---|---|---|---|---|---|
pn# + 1 [14] | pn# − 1 [15] | ||||
0 | 1 | — | 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 |
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures.
10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.
A highly composite number is a positive integer with more divisors than any smaller positive integer has. A related concept is that of a largely composite number, a positive integer which has at least as many divisors as any smaller positive integer. The name can be somewhat misleading, as the first two highly composite numbers are not actually composite numbers; however, all further terms are.
The tables contain the prime factorization of the natural numbers from 1 to 1000.
In number theory, Bertrand's postulate is the theorem that for any integer , there exists at least one prime number with
31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.
104 is the natural number following 103 and preceding 105.
120 is the natural number following 119 and preceding 121.
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360 is the natural number following 359 and preceding 361.
2000 is a natural number following 1999 and preceding 2001.
A highly totient number is an integer that has more solutions to the equation , where is Euler's totient function, than any integer below it. The first few highly totient numbers are
In number theory, a practical number or panarithmic number is a positive integer such that all smaller positive integers can be represented as sums of distinct divisors of . For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2.
100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.
In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power.
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In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev functionϑ (x) or θ (x) is given by
288 is the natural number following 287 and preceding 289. Because 288 = 2 · 12 · 12, it may also be called "two gross" or "two dozen dozen".
744 is the natural number following 743 and preceding 745.