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A211773 Prime-generating polynomial: a(n) = 2*n^2 - 108*n + 1259.

%I A211773 #83 Feb 20 2025 14:02:40
%S A211773 1259,1153,1051,953,859,769,683,601,523,449,379,313,251,193,139,89,43,
%T A211773 1,-37,-71,-101,-127,-149,-167,-181,-191,-197,-199,-197,-191,-181,
%U A211773 -167,-149,-127,-101,-71,-37,1,43,89,139,193,251,313,379,449,523,601,683,769,859,953
%N A211773 Prime-generating polynomial: a(n) = 2*n^2 - 108*n + 1259.
%C A211773 This polynomial generates 92 primes (66 distinct ones) for 0 <= n <= 99 (in fact the next two terms are still primes but we keep the range 0-99, customary for comparisons), just three primes less than the record held by Euler's polynomial for n = m - 35, which is m^2 - 69*m + 1231 (see the link below), but having six distinct primes more than this one.
%C A211773 The nonprime terms in the first 100 are: 1 (taken twice), 1369 = 37^2, 1849 = 43^2, 4033 = 37*109, 5633 = 43*131, 7739 = 71*109 and 8251 = 37*223.
%C A211773 For n = 2*m - 34 we obtain the polynomial 8*m^2 - 488*m + 7243, which generates 31 primes in a row starting from m = 0 (polynomial already reported, see the link below).
%C A211773 For n = 4*m - 34 we obtain the polynomial 32*m^2 - 976*m + 7243, which generates 31 primes in row starting from m = 0.
%C A211773 The polynomial 2*n^2 + 40*n + 1, which generates the positive terms of this sequence in ascending order (i.e., a(37), ...), yields 10774009 distinct primes for 0 <= n < 49999999 while Euler's polynomial (n^2 - n + 41) gives 9967520 primes in same range. - _Mikk Heidemaa_, Feb 23 2016
%D A211773 Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials in Lecture Notes in Pure and Applied Mathematics (Vol. 220), Marcel Dekker Inc., 2001, pages 281-317.
%H A211773 Bruno Berselli, <a href="/A211773/b211773.txt">Table of n, a(n) for n = 0..1000</a>
%H A211773 Marius Coman, <a href="https://www.researchgate.net/publication/277912540">Ten prime-generating quadratic polynomials</a>, Preprint 2015.
%H A211773 Joe L. Mott and Kermite Rose, <a href="http://www.math.fsu.edu/~aluffi/archive/paper134.ps">Prime-Producing Cubic Polynomials</a>.
%H A211773 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.
%H A211773 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A211773 G.f.: (1259 - 2624*x + 1369*x^2)/(1-x)^3. - _Bruno Berselli_, May 18 2012
%F A211773 a(n-37) = 2*n^2 + 40*n + 1. - _Mikk Heidemaa_, Feb 18 2016
%F A211773 From _Elmo R. Oliveira_, Feb 09 2025: (Start)
%F A211773 E.g.f.: exp(x)*(1259 - 106*x + 2*x^2).
%F A211773 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A211773 Table[2 n^2 + 40 n + 1, {n, -37, 962}] (* _Mikk Heidemaa_, Feb 18 2016 *)
%o A211773 (Magma) [2*n^2-108*n+1259: n in [0..49]]; // _Bruno Berselli_, May 18 2012
%o A211773 (PARI) a(n)=2*n^2 - 108*n + 1259 \\ _Charles R Greathouse IV_, Jun 29 2017
%Y A211773 Cf. A060566, A202018.
%K A211773 sign,easy
%O A211773 0,1
%A A211773 _Marius Coman_, May 18 2012