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A155557 A proximate-prime polynomial sequence generated by 2n^2 - 2n + 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 2n^2 + 24n + 144323 generates more primes than 2n^2 - 2n + 53089. - Mike Winkler, Oct 25 2013.

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

Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

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]

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

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 July 22 16:32 EDT 2017. Contains 289671 sequences.