Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #76 Sep 08 2022 08:46:02
%S 1259,1153,1051,953,859,769,683,601,523,449,379,313,251,193,139,89,43,
%T 1,-37,-71,-101,-127,-149,-167,-181,-191,-197,-199,-197,-191,-181,
%U -167,-149,-127,-101,-71,-37,1,43,89,139,193,251,313,379,449,523,601,683,769
%N Prime-generating polynomial: 2*n^2 - 108*n + 1259.
%C This polynomial generates 92 primes (66 distinct ones) for n from 0 to 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 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 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 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 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 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 Bruno Berselli, <a href="/A211773/b211773.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Coman, <a href="https://www.researchgate.net/publication/277912540">Ten prime-generating quadratic polynomials</a>, Preprint 2015.
%H Joe L. Mott and Kermite Rose, <a href="http://www.math.fsu.edu/~aluffi/archive/paper134.ps">Prime-Producing Cubic Polynomials</a>
%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">MathWorld: Prime-Generating Polynomial</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: (1259-2624*x+1369*x^2)/(1-x)^3. - _Bruno Berselli_, May 18 2012
%F a(n-37) = 2*n^2 + 40*n + 1. - _Mikk Heidemaa_, Feb 18 2016
%t Table[2 n^2 + 40 n + 1, {n, -37, 962}] (* _Mikk Heidemaa_, Feb 18 2016 *)
%o (Magma) [2*n^2-108*n+1259: n in [0..49]]; // _Bruno Berselli_, May 18 2012
%o (PARI) a(n)=2*n^2 - 108*n + 1259 \\ _Charles R Greathouse IV_, Jun 29 2017
%Y Cf. A060566, A202018.
%K sign,easy
%O 0,1
%A _Marius Coman_, May 18 2012