This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A213810 #48 Feb 09 2025 11:54:50
%S A213810 14561,14083,13613,13151,12697,12251,11813,11383,10961,10547,10141,
%T A213810 9743,9353,8971,8597,8231,7873,7523,7181,6847,6521,6203,5893,5591,
%U A213810 5297,5011,4733,4463,4201,3947,3701,3463,3233,3011,2797,2591,2393,2203,2021,1847,1681,1523
%N A213810 a(n) = 4*n^2 - 482*n + 14561.
%C A213810 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).
%C A213810 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.
%C A213810 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).
%D A213810 W. Narkiewicz, The Development of Prime Number Theory: from Euclid to Hardy and Littlewood, Springer Monographs in Mathematics, 2000, page 43.
%H A213810 G. C. Greubel, <a href="/A213810/b213810.txt">Table of n, a(n) for n = 0..1000</a>
%H A213810 R. A. Mollin, <a href="http://www.jstor.org/stable/2975080">Prime-Producing Quadratics</a>, The American Mathematical Monthly, Vol. 104, No. 6 (1997), pp. 529-544.
%H A213810 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_232.htm">Puzzle 232: Primes and Cubic polynomials</a>, The Prime Puzzles & Problems Connection.
%H A213810 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A213810 a(n) = 4*n^2 - 482*n + 14561.
%F A213810 G.f.: (-15047*x^2 + 29600*x - 14561)/(x-1)^3. - _Alexander R. Povolotsky_, Jun 21 2012
%F A213810 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _G. C. Greubel_, Feb 26 2017
%F A213810 E.g.f.: exp(x)*(14561 - 478*x + 4*x^2). - _Elmo R. Oliveira_, Feb 09 2025
%t A213810 Table[4n^2-482n+14561,{n,0,41}] (* _Harvey P. Dale_, Sep 09 2014 *)
%t A213810 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 *)
%o A213810 (PARI) x='x+O('x^50); Vec((-15047*x^2+29600*x-14561)/(x-1)^3) \\ _G. C. Greubel_, Feb 26 2017
%Y A213810 Cf. A181973, A211773, A211775.
%K A213810 nonn,easy
%O A213810 0,1
%A A213810 _Marius Coman_, Jun 20 2012
%E A213810 Edited by _N. J. A. Sloane_, Nov 12 2016