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%I #42 Nov 05 2019 05:55:15
%S 14561,14083,13613,13151,12697,12251,11813,11383,10961,10547,10141,
%T 9743,9353,8971,8597,8231,7873,7523,7181,6847,6521,6203,5893,5591,
%U 5297,5011,4733,4463,4201,3947,3701,3463,3233,3011,2797,2591,2393,2203,2021,1847,1681,1523
%N a(n) = 4*n^2 - 482*n + 14561.
%C A "prime-generating" polynomial: This polynomial generates 88 distinct primes for n from 0 to 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 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 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 W. Narkiewicz, The Development of Prime Number Theory: from Euclid to Hardy and Littlewood, Springer Monographs in Mathematics, 2000, page 43.
%H G. C. Greubel, <a href="/A213810/b213810.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 4*n^2 - 482*n + 14561.
%F G.f.: (-15047*x^2+29600*x-14561)/(x-1)^3. - _Alexander R. Povolotsky_, Jun 21 2012
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _G. C. Greubel_, Feb 26 2017
%t Table[4n^2-482n+14561,{n,0,41}] (* _Harvey P. Dale_, Sep 09 2014 *)
%t 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 (PARI) x='x+O('x^50); Vec((-15047*x^2+29600*x-14561)/(x-1)^3) \\ _G. C. Greubel_, Feb 26 2017
%Y Cf. A181973, A211773, A211775.
%K nonn,easy
%O 0,1
%A _Marius Coman_, Jun 20 2012
%E Edited by _N. J. A. Sloane_, Nov 12 2016
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