A155557 A proximate-prime polynomial sequence generated by 2*n^2 - 2*n + 53089.

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, 54949, 55073, 55201, 55333, 55469, 55609, 55753, 55901

Offset: 1

Author

Michael M. Ross, Jan 24 2009

Keywords

• easy
• nonn

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

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Natural Numbers, The High Primality of Prime-Derived Quadratic Sequences.
• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A126665, A126719, A127316, A140947.

Programs

Magma

[2*n^2 - 2*n + 53089: n in [1..35]]; // Vincenzo Librandi, Jul 20 2011

Mathematica

Table[2n^2-2n+53089,{n,30}] (* or *) LinearRecurrence[{3,-3,1},{53089,53093,53101},30] (* Harvey P. Dale, Jul 19 2011 *)

Other

QTest: Derive, analyze and solve quadratic expressions. Generate integer sequences and determine their primality. (http://www.naturalnumbers.org/QTest-NTK.html)

PARI

a(n)=2*n^2-2*n+53089 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 2*n^2 - 2*n + 53089.

From Harvey P. Dale, Jul 19 2011: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4.

G.f.: x*(53089 - 106174*x + 53089*x^2)/(1-x)^3. (End)

E.g.f.: exp(x)*(2*x^2 + 53089) - 53089. - Elmo R. Oliveira, Nov 09 2024

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010