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| {{for|the computer program|SuperPrime}}
| | Hi there, I am Alyson Boon even though it is not the title on my beginning certification. Mississippi is where her house is but her husband desires them to transfer. To climb is something I truly appreciate performing. Office supervising is what she does for a living.<br><br>Visit my blog: clairvoyance - [http://Ustanford.com/index.php?do=/profile-38218/info/ http://Ustanford.com/index.php?do=/profile-38218/info/], |
| '''Super-prime numbers''' (also known as "'''higher order primes'''") are the [[subsequence]] of [[prime numbers]] that occupy prime-numbered positions within the sequence of all prime numbers. The subsequence begins
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| :3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, … {{OEIS|id=A006450}}.
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| That is, if ''p''(''i'') denotes the ''i''th prime number, the numbers in this sequence are those of the form ''p''(''p''(''i'')). {{harvtxt|Dressler|Parker|1975}} used a computer-aided proof (based on calculations involving the [[subset sum problem]]) to show that every integer greater than 96 may be represented as a sum of distinct super-prime numbers. Their proof relies on a result resembling [[Bertrand's postulate]], stating that (after the larger gap between super-primes 5 and 11) each super-prime number is less than twice its predecessor in the sequence.
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| Broughan and Barnett<ref>Kevin A. Broughan and A. Ross Barnett, [http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Broughan/broughan16.html On the Subsequence of Primes Having Prime Subscripts], ''Journal of Integer Sequences'' '''12''' (2009), article 09.2.3.</ref> show that there are
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| :<math>\frac{x}{(\log x)^2}+O\left(\frac{x\log\log x}{(\log x)^3}\right)</math> | |
| super-primes up to ''x''.
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| This can be used to show that the set of all super-primes is [[Small set (combinatorics)|small]].
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| One can also define "higher-order" primeness much the same way, and obtain analogous sequences of primes. {{harvtxt|Fernandez|1999}}
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| A variation on this theme is the sequence of prime numbers with [[palindromic number|palindromic]] indices, beginning with
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| :3, 5, 11, 17, 31, 547, 739, 877, 1087, 1153, 2081, 2381, … {{OEIS|id=A124173}}. | |
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| ==References==
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| <references/>
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| *{{citation
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| | first1 = Robert E. | last1 = Dressler
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| | first2 = S. Thomas | last2 = Parker
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| | title = Primes with a prime subscript
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| | journal = Journal of the ACM
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| | volume = 22
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| | issue = 3
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| | year = 1975
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| | pages = 380–381
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| | doi = 10.1145/321892.321900
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| | mr = 0376599}}.
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| *{{citation
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| | first1 =Neil | last1 = Fernandez
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| | title = An order of primeness, F(p)
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| | url = http://borve.org/primeness/FOP.html
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| | year = 1999}}.
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| ==External links==
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| *[http://acm.sgu.ru/problem.php?contest=0&problem=116 A Russian programming contest problem related to the work of Dressler and Parker]
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| {{Prime number classes}}
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| [[Category:Classes of prime numbers]]
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| {{numtheory-stub}}
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Hi there, I am Alyson Boon even though it is not the title on my beginning certification. Mississippi is where her house is but her husband desires them to transfer. To climb is something I truly appreciate performing. Office supervising is what she does for a living.
Visit my blog: clairvoyance - http://Ustanford.com/index.php?do=/profile-38218/info/,