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A155557
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A proximate-prime polynomial sequence generated by 2n^2 - 2n + 53089. Produces 634 primes in the first 1000 terms. (A proximate-prime polynomial is a finite polynomial equation that is derived from four successive - proximate, or neighboring - primes.)
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53089, 53093, 53101, 53113, 53129, 53149, 53173, 53201, 53233, 53269, 53309, 53353, 53401, 53453, 53509, 53569, 53633, 53701, 53773, 53849, 53929, 54013, 54101, 54193, 54289, 54389, 54493, 54601, 54713, 54829
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Quadratic derived from four successive primes: 53089, 53093, 53101, 53113. Produces more primes in the first 1000 terms than any other quadratic derived from 4 successive primes under 1000000. (This includes 41, 43, 47, 53 = n^2 - n + 41, which produces 582.)
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..10000
The High Primality of Prime-Derived Quadratic Sequences
Wolfram Mathworld: Prime-Generating Polynomial
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FORMULA
| a(n) = 2*n^2 - 2*n + 53089.
a(1)=53089, a(2)=53093, a(3)=53101, a(n)=3*a(n-1)-3*a(n-2)+a(n-3) [From Harvey P. Dale, Jul 19 2011]
G.f.: x*(-53089+106174*x-53089*x^2)/(x-1)^3 [From Harvey P. Dale, Jul 19 2011]
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EXAMPLE
| For n=14, 2*(14^2)-(2*14)+53089 = 53453
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MATHEMATICA
| Table[2n^2-2n+53089, {n, 30}] (* or *) LinearRecurrence[{3, -3, 1}, {53089, 53093, 53101}, 30] (* From Harvey P. Dale, Jul 19 2011 *)
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PROG
| (Other) QTest: Derive, analyze and solve quadratic expressions. Generate integer sequences and determine their primality. (http://www.naturalnumbers.org/QTest-NTK.html)
(MAGMA) [2*n^2 - 2*n + 53089: n in [1..35]]; // Vincenzo Librandi, Jul 20 2011
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CROSSREFS
| A140947, A126665, A126719, A127316
Sequence in context: A206092 A157758 A031833 * A015315 A061330 A195656
Adjacent sequences: A155554 A155555 A155556 * A155558 A155559 A155560
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KEYWORD
| easy,nonn
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AUTHOR
| Michael M. Ross (michaelmross(AT)gmail.com), Jan 24 2009
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EXTENSIONS
| Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Jul 25 2010
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