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%I #30 Sep 08 2022 08:46:21
%S 1697,4027,6359,8693,11029,13367,15707,18049,20393,22739,25087,27437,
%T 29789,32143,34499,36857,39217,41579,43943,46309,48677,51047,53419,
%U 55793,58169,60547,62927,65309,67693,70079,72467,74857,77249,79643,82039,84437,86837,89239,91643,94049,96457,98867,101279
%N a(n) = n^2 + 2329n + 1697.
%C This quadratic seems to be good at generating many distinct primes. It generates 642 primes for n < 10^3, 5132 primes for n < 10^4, 41224 primes for n < 10^5 and 340009 primes for n < 10^6. The first few primes generated are 1697, 4027, 6359, 8693, 13367, 18049, 20393, 22739, 25087, 27437.
%C The quadratic was first discovered as b(n) = n^2 + 1151n - 1023163 by Steve Trevorrow in 2006 during Al Zimmermann's Prime Generating Polynomials contest (see link). Note that a(n) = b(n + 589).
%C Smallest n such that a(n) is not squarefree is 704; a(704) = 2136929 = 73^2*401 and smallest n such that a(n) is square is 1327; a(1327) = 4853209 = 2203^2. - _Altug Alkan_, Mar 30 2018
%H Colin Barker, <a href="/A301985/b301985.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%H Al Zimmermann, <a href="http://recmath.com/contest/PGP/index.php">Prime Generating Polynomials contest</a>, 2006.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 2a(n-1) - a(n-2) + 2, a(1) = 1697, a(2) = 4027.
%F From _Colin Barker_, Mar 30 2018: (Start)
%F G.f.: (1697 - 1064*x - 631*x^2) / (1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
%F (End)
%t Table[n^2 + 2329 n + 1697, {n, 0, 50}] (* _Vincenzo Librandi_, Mar 31 2018 *)
%o (PARI) a(n) = n^2 + 2329*n + 1697; \\ _Altug Alkan_, Mar 30 2018
%o (PARI) Vec((1697 - 1064*x - 631*x^2) / (1 - x)^3 + O(x^60)) \\ _Colin Barker_, Mar 30 2018
%o (Magma) [n^2+2329*n+1697: n in [0..50]]; // _Vincenzo Librandi_, Mar 31 2018
%Y Cf. A005846, A050267, A050268.
%K nonn,easy
%O 0,1
%A _Dmitry Kamenetsky_, Mar 30 2018
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