A213810 a(n) = 4*n^2 - 482*n + 14561.

14561, 14083, 13613, 13151, 12697, 12251, 11813, 11383, 10961, 10547, 10141, 9743, 9353, 8971, 8597, 8231, 7873, 7523, 7181, 6847, 6521, 6203, 5893, 5591, 5297, 5011, 4733, 4463, 4201, 3947, 3701, 3463, 3233, 3011, 2797, 2591, 2393, 2203, 2021, 1847, 1681, 1523

Offset: 0

Author

Marius Coman, Jun 20 2012

Keywords

• nonn
• easy

Comments

A "prime-generating" polynomial: This polynomial generates 88 distinct primes for 0 <= n <= 99, just two primes fewer than the record held by the polynomial discovered by N. Boston and M. L. Greenwood, that is 41*n^2 - 4641*n + 88007 (this polynomial is sometimes cited as 41*n^2 + 33*n - 43321, which is the same for the input values [-57,42], see the references below).

The nonprime terms in the first 100 are: 10961 = 97*113; 10547 = 53*199; 9353 = 47*199; 7181 = 43*167; 6847 = 41*167; 5893 = 71*83; 3233 = 53*61; 2021 = 43*47; 1681 = 41^2; 1763 = 41*43; 2491 = 47*53; 4331 = 61*71.

For n = m + 41 we obtain the polynomial 4*m^2 - 154*m + 1523, which generates 40 primes in a row starting from m = 0 (polynomial already reported, see the link below).

References

• W. Narkiewicz, The Development of Prime Number Theory: from Euclid to Hardy and Littlewood, Springer Monographs in Mathematics, 2000, page 43.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (1997), pp. 529-544.
• Carlos Rivera, Puzzle 232: Primes and Cubic polynomials, The Prime Puzzles & Problems Connection.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A181973, A211773, A211775.

Programs

Mathematica

Table[4n^2-482n+14561,{n,0,41}] (* Harvey P. Dale, Sep 09 2014 *)
LinearRecurrence[{3,-3,1},{14561, 14083, 13613}, 50] (* or *) CoefficientList[Series[ (-15047*x^2+29600*x-14561)/(x-1)^3, {x,0,50}], x] (* G. C. Greubel, Feb 26 2017 *)

PARI

x='x+O('x^50); Vec((-15047*x^2+29600*x-14561)/(x-1)^3) \\ G. C. Greubel, Feb 26 2017

Formula

a(n) = 4*n^2 - 482*n + 14561.

G.f.: (-15047*x^2 + 29600*x - 14561)/(x-1)^3. - Alexander R. Povolotsky, Jun 21 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - G. C. Greubel, Feb 26 2017

E.g.f.: exp(x)*(14561 - 478*x + 4*x^2). - Elmo R. Oliveira, Feb 09 2025

Extensions

Edited by N. J. A. Sloane, Nov 12 2016