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
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).
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
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
KEYWORD
easy,nonn
AUTHOR
Michael M. Ross, Jan 24 2009
EXTENSIONS
Edited by Charles R Greathouse IV, Jul 25 2010
STATUS
approved