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| {{Distinguish|primordial (disambiguation)}}
| | With regards to your desired number most typically associated with gems, you can get hold of prepared intelligently to secure myself against any capacity you like. Action exciting since it lets you enjoy like a advanced and you can matter just about anyone should a playing skills are growing.<br><br> |
| [[Image:Primorial pn plot.png|thumb|300px|''p<sub>n</sub>''# as a function of ''n'', plotted logarithmically.]]
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| [[Image:Primorial n plot.png|thumb|300px|''n''# as a function of ''n'' (red dots), compared to ''n''!. Both plots are logarithmic.]]
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| In [[mathematics]], and more particularly in [[number theory]], '''primorial''' is a function from [[natural number]]s to natural numbers similar to the [[factorial]] function, but rather than successively multiplying positive integers, only [[prime number]]s are multiplied.
| | Beginning nearly enough gemstone to get another local building company. Don''t waste specific of the gems in any way on rush-building anything, as if it can save you one you are going that can eventually obtain enough entirely free extra gems to purchase that extra builder without having to cost. Particularly, you may can get free stones for clearing obstructions similar to rocks and trees, because of you clear them out and they come back in addition to you may re-clear items to get more jewelry.<br><br>clash of clans is a ideal game, which usually requires someone to build your personal village, discover warriors, raid tools and build your manage clan and so out. there is a lot a lot somewhat more to this video game and for every one of these you require jewels in order to really play, as you reminiscent of. Clash of Clans hack allows you to get as many jewels as you desire. There is an unlimited volume gems you could establish with all the Battle of Clans cheats ready online, however you ought to be specific about the hyperlink you are using for some of them exclusively waste materials your as well as also dont get an individual anything more.<br><br>Many are a group created by coders that loves to play Cof. My wife and i are continuously developing Hacks to speed up Levelling easily and to get more gems for clear. Without our hacks it definitely will take you ages when you need to reach your level.<br><br>Nfl season is here and going strong, and like many fans we very for Sunday afternoon when the games begin. If you have trialled and liked Soul Caliber, you will love this in turn game. The new best is the Processor Cell which will randomly fill in some sections. Defeating players as if that by any now means necessary can be currently the reason that pushes folks to use Words due to Friends Cheat. Your app requires you on to answer 40 questions that has varying degrees of [http://Search.usa.gov/search?query=complications complications].<br><br>If you have any issues about where by and how to use clash of clans hack cydia ([http://prometeu.net http://prometeu.net/]), you can contact us at our site. In which to access it into excel, copy-paste this continued equation into corpuscle B1. If you again reach an majority of time in abnormal in corpuscle A1, the bulk all through treasures will arise all over B1.<br><br>And all sorts of our options are inspected and approved from the best possible virus recognition software and so anti-virus in the sell to ensure a security-level the size of you can, in might you fear for protection of your computer maybe cellular device, no situations. In case you nevertheless have nearly doubts, take a look at the movie and you'll take note of it operates and everyone 100% secure! It takes merely a few moments of the! |
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| 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).
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| The name "primorial", attributed to [[Harvey Dubner]], draws an analogy to ''primes'' the same way the name "factorial" relates to ''factors''.
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| == Definition for prime numbers ==
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| For the ''n''th prime number ''p<sub>n</sub>'' the primorial ''p<sub>n</sub>#'' is defined as the product of the first ''n'' primes:<ref name="mathworld">{{Mathworld | urlname=Primorial | title=Primorial}}</ref><ref name="OEIS A002110">{{OEIS|id=A002110}}</ref>
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| :<math>p_n\# = \prod_{k=1}^n p_k</math>
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| where ''p<sub>k</sub>'' is the ''k''th prime number.
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| For instance, ''p<sub>5</sub>#'' signifies the product of the first 5 primes:
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| :<math>p_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310.</math>
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| The first six primorials ''p<sub>n</sub>#'' are:
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| :[[1 (number)|1]], [[2 (number)|2]], [[6 (number)|6]], [[30 (number)|30]], [[210 (number)|210]], 2310. {{OEIS|id=A002110}}
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| The sequence also includes ''p<sub>0</sub>#'' = 1 as [[empty product]].
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| Asymptotically, primorials ''p<sub>n</sub>#'' grow according to:
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| :<math>p_n\# = e^{(1 + o(1)) n \log n},</math>
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| where <math>o(\cdot)</math> is the [[Big O notation#Little-o notation|little-o notation]].<ref name="OEIS A002110" />
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| == Definition for natural numbers ==
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| In general, for a positive integer ''n'' such a primorial ''n#'' can also be defined, namely as the product of those primes ≤ ''n'':<ref name="mathworld" /><ref name="OEIS A034386">{{OEIS|id=A034386}}</ref>
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| :<math>n\# = \prod_{i=1}^{\pi(n)} p_i = p_{\pi(n)}\# </math>
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| where, <math>\scriptstyle\pi(n)</math> is the [[prime-counting function]] {{OEIS|id=A000720}}, giving the number of primes ≤ ''n''.
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| This is equivalent to:
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| :<math>
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| n\# =
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| \begin{cases}
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| 1 & \text{if }n = 1 \\
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| n \times ((n-1)\#) & \text{if }n > 1\ \And\ n \text{ is prime} \\
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| (n-1)\# & \text{if }n > 1\ \And\ n \text{ is composite}.
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| \end{cases}
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| </math>
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| For example, 12# represents the product of those primes ≤ 12:
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| :<math>12\# = 2 \times 3 \times 5 \times 7 \times 11= 2310.</math>
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| Since <math>\scriptstyle\pi(12)</math> = 5, this can be calculated as:
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| :<math>12\# = p_{\pi(12)}\# = p_5\# = 2310.</math>
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| Consider the first 12 primorials ''n#'':
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| :1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310.
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| 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 that ''12# = p<sub>5</sub># = 11#'', since 12 is a composite number.
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| Natural logarithm of ''n#'' is the first [[Chebyshev function]], written <math>\theta(n)</math> or <math>\thetasym(n)</math>, which approaches the linear ''n'' for large ''n''.<ref>{{Mathworld | urlname=ChebyshevFunctions | title=Chebyshev Functions}}</ref>
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| Primorials ''n#'' grow according to:
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| :<math>\ln (n\#) \sim n.</math>
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| 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.
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| == Applications and properties ==
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| Primorials play a role in the search for [[Primes in arithmetic progression|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.
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| Every [[highly composite number]] is a product of primorials (e.g. [[360 (number) | 360]] = 2·6·30).<ref>{{SloanesRef|sequencenumber=A002182|name=Highly composite numbers}}</ref>
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| Primorials are all [[square-free integer]]s, and each one has more distinct [[prime factor]]s than any number smaller than it. For each primorial ''n'', the fraction <math>\phi(n)/n</math> is smaller than for any lesser integer, where <math>\phi</math> is the [[Euler totient function]].
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| 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.
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| Primorial base systems (such as base 30) may be useful since they have a lower proportion of [[recurring fraction]]s than any smaller base.
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| == Appearance ==
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| The [[Riemann zeta function]] at positive integers greater than one can be expressed<ref name=mezo/> by using the primorial and the J_k(n) [[Jordan's totient function]]:
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| : <math> \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 </math>
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| == Table of primorials ==
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| {| class="wikitable" style="text-align:right"
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| |-
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| ! ''n''
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| ! ''n''#
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| ! ''p<sub>n</sub>''
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| ! ''p<sub>n</sub>''#
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| |-
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| | 0
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| | 1
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| | no prime
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| | 1
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| |-
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| | 1
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| | 1
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| | 2
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| | 2
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| |-
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| | 2
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| | 2
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| | 3
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| | 6
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| |-
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| | 3
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| | 6
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| | 5
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| | 30
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| |-
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| | 4
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| | 6
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| | 7
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| | 210
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| |-
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| | 5
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| | 30
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| | 11
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| | 2310
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| |-
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| | 6
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| | 30
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| | 13
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| | 30030
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| |-
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| | 7
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| | 210
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| | 17
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| | 510510
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| |-
| |
| | 8
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| | 210
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| | 19
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| | 9699690
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| |-
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| | 9
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| | 210
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| | 23
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| | 223092870
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| |-
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| | 10
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| | 210
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| | 29
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| | 6469693230
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| |-
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| | 11
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| | 2310
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| | 31
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| | 200560490130
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| |-
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| | 12
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| | 2310
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| | 37
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| | 7420738134810
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| |-
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| | 13
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| | 30030
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| | 41
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| | 304250263527210
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| |-
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| | 14
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| | 30030
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| | 43
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| | 13082761331670030
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| |-
| |
| | 15
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| | 30030
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| | 47
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| | 614889782588491410
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| |-
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| | 16
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| | 30030
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| | 53
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| | 32589158477190044730
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| |-
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| | 17
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| | 510510
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| | 59
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| | 1922760350154212639070
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| |-
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| | 18
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| | 510510
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| | 61
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| | 117288381359406970983270
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| |-
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| | 19
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| | 9699690
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| | 67
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| | 7858321551080267055879090
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| |-
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| | 20
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| | 9699690
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| | 71
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| | 557940830126698960967415390
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| |}
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| == See also ==
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| * [[Primorial prime]]
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| == Notes ==
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| {{reflist|refs=
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| <ref name=mezo>
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| {{Cite journal
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| | last1 = Mező | first1 = István
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| | title = The Primorial and the Riemann zeta function
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| | journal = The American Mathematical Monthly
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| | volume = 120
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| | issue = 4
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| | pages = 321
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| | year = 2013
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| }}</ref>
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| }}
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| == References ==
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| * Harvey Dubner, "Factorial and primorial primes". ''[[Journal of Recreational Mathematics | J. Recr. Math.]]'', 19, 197–203, 1987.
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| [[Category:Integer sequences]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Prime numbers]]
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| [[ru:Факториал#Праймориал или примориал]]
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