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%I #33 Mar 06 2023 08:54:07
%S 53089,53093,53101,53113,53129,53149,53173,53201,53233,53269,53309,
%T 53353,53401,53453,53509,53569,53633,53701,53773,53849,53929,54013,
%U 54101,54193,54289,54389,54493,54601,54713,54829
%N A proximate-prime polynomial sequence generated by 2*n^2 - 2*n + 53089.
%C Sequence 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.)
%C 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.)
%C For larger ranges of n, for example n=0..10^6 or n=0..10^7, the polynomial 2*n^2 + 24*n + 144323 generates more primes than 2*n^2 - 2*n + 53089. - _Mike Winkler_, Oct 25 2013
%H Vincenzo Librandi, <a href="/A155557/b155557.txt">Table of n, a(n) for n = 1..10000</a>
%H Natural Numbers, <a href="http://www.naturalnumbers.org/ppanalysis.html">The High Primality of Prime-Derived Quadratic Sequences</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 2*n^2 - 2*n + 53089.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=53089, a(2)=53093, a(3)=53101. - _Harvey P. Dale_, Jul 19 2011
%F G.f.: x*(-53089 + 106174*x - 53089*x^2)/(x-1)^3. - _Harvey P. Dale_, Jul 19 2011
%e For n=14, 2*(14^2) - (2*14) + 53089 = 53453.
%t Table[2n^2-2n+53089,{n,30}] (* or *) LinearRecurrence[{3,-3,1},{53089,53093,53101},30] (* _Harvey P. Dale_, Jul 19 2011 *)
%o (Other) QTest: Derive, analyze and solve quadratic expressions. Generate integer sequences and determine their primality. (http://www.naturalnumbers.org/QTest-NTK.html)
%o (Magma) [2*n^2 - 2*n + 53089: n in [1..35]]; // _Vincenzo Librandi_, Jul 20 2011
%o (PARI) a(n)=2*n^2-2*n+53089 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A140947, A126665, A126719, A127316.
%K easy,nonn
%O 1,1
%A _Michael M. Ross_, Jan 24 2009
%E Edited by _Charles R Greathouse IV_, Jul 25 2010
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