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 A155557 A proximate-prime polynomial sequence generated by 2*n^2 - 2*n + 53089. 1
 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; text; internal format)
 OFFSET 1,1 COMMENTS 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.) 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.) 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..10000 Index entries for linear recurrences with constant coefficients, signature (3, -3, 1). FORMULA a(n) = 2*n^2 - 2*n + 53089. 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 G.f.: x*(-53089 + 106174*x - 53089*x^2)/(x-1)^3. - Harvey P. Dale, Jul 19 2011 EXAMPLE For n=14, 2*(14^2) - (2*14) + 53089 = 53453. MATHEMATICA Table[2n^2-2n+53089, {n, 30}] (* or *) LinearRecurrence[{3, -3, 1}, {53089, 53093, 53101}, 30] (* Harvey P. Dale, Jul 19 2011 *) 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 (PARI) a(n)=2*n^2-2*n+53089 \\ Charles R Greathouse IV, Jun 17 2017 CROSSREFS Cf. A140947, A126665, A126719, A127316. Sequence in context: A250696 A250681 A031833 * A015315 A061330 A195656 Adjacent sequences:  A155554 A155555 A155556 * A155558 A155559 A155560 KEYWORD easy,nonn AUTHOR Michael M. Ross, Jan 24 2009 EXTENSIONS Edited by Charles R Greathouse IV, Jul 25 2010 STATUS approved

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Last modified May 21 22:24 EDT 2019. Contains 323467 sequences. (Running on oeis4.)