

A155557


A proximateprime 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
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OFFSET

1,1


COMMENTS

Sequence produces 634 primes in the first 1000 terms. (A proximateprime 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
The High Primality of PrimeDerived Quadratic Sequences
Wolfram Mathworld: PrimeGenerating Polynomial
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(n1)  3*a(n2) + a(n3); a(1)=53089, a(2)=53093, a(3)=53101.  Harvey P. Dale, Jul 19 2011
G.f.: x*(53089 + 106174*x  53089*x^2)/(x1)^3.  Harvey P. Dale, Jul 19 2011


EXAMPLE

For n=14, 2*(14^2)  (2*14) + 53089 = 53453.


MATHEMATICA

Table[2n^22n+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/QTestNTK.html)
(MAGMA) [2*n^2  2*n + 53089: n in [1..35]]; // Vincenzo Librandi, Jul 20 2011
(PARI) a(n)=2*n^22*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



