Primorial

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Categories: Integer sequences | Factorial and binomial topics

n# as a function of n (red dots), compared to n!. Both plots are logarithmic.

p_n# as a function of n, plotted logarithmically.

For n ≥ 2, the primorial (n#) is the product of all prime numbers less than or equal to n. For example, 7# = 210 is a primorial which is the product of the first four primes multiplied together (2·3·5·7). The name is attributed to Harvey Dubner and is a portmanteau of prime and factorial. The first few primorials are:

2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, ... (sequence A002110 in OEIS)

Notation varies; it's common to see p_n#, indicating the product of the primes less than or equal to the nth prime (in other words, the product of the first n primes), and also a(n) = p_n#. Asymptotically, primorials grow according to:

Failed to parse (Missing texvc executable; please see math/README to configure.): p_n\# = \exp\left((1 + o(1)) \cdot n \log n\right),

Failed to parse (Missing texvc executable; please see math/README to configure.): \log n\# \sim n.

where "exp" is the exponential function e^x and "o" is the "little-o" notation (see Big O notation).^[1] Its natural logarithm is the first Chebyshev function, written Failed to parse (Missing texvc executable; please see math/README to configure.): \theta(n)

or Failed to parse (Missing texvc executable; please see math/README to configure.): \thetasym(n)

, which approaches the linear n for large n.^[2]

The idea of multiplying all primes occurs in a proof of the infinitude of the prime numbers; it is applied to show a contradiction in the idea that the primes could be finite in number.

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).

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \phi(n)/n

is smaller than for any lesser integer, where Failed to parse (Missing texvc executable; please see math/README to configure.): \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.

1 Table of primorials
2 See also
3 References
4 External links

Table of primorials

p	p#	No. of Digits
2	2	1
3	6	1
5	30	2
7	210	3
11	2310	4
13	30030	5
17	510510	6
19	9699690	7
23	223092870	9
29	6469693230	10
31	200560490130	12
37	7420738134810	13
41	304250263527210	15
43	13082761331670030	17
47	614889782588491410	18
53	32589158477190044730	20
59	1922760350154212639070	22
61	117288381359406970983270	24
67	7858321551080267055879090	25
71	557940830126698960967415390	27
73	40729680599249024150621323470	29
79	3217644767340672907899084554130	31
83	267064515689275851355624017992790	33
89	23768741896345550770650537601358310	35
97	2305567963945518424753102147331756070	37
101	232862364358497360900063316880507363070	39
103	23984823528925228172706521638692258396210	41
107	2566376117594999414479597815340071648394470	43
109	279734996817854936178276161872067809674997230	45
113	31610054640417607788145206291543662493274686990	47