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A181973 Prime-generating polynomial: a(n) = 16*n^2 - 300*n + 1447.

%I A181973 #51 Feb 09 2025 14:21:50
%S A181973 1447,1163,911,691,503,347,223,131,71,43,47,83,151,251,383,547,743,
%T A181973 971,1231,1523,1847,2203,2591,3011,3463,3947,4463,5011,5591,6203,6847,
%U A181973 7523,8231,8971,9743,10547,11383,12251,13151,14083,15047,16043,17071,18131,19223,20347
%N A181973 Prime-generating polynomial: a(n) = 16*n^2 - 300*n + 1447.
%C A181973 This polynomial generates 30 primes in a row starting from n = 0.
%C A181973 The polynomial 16*n^2 - 628*n + 6203 generates the same primes in reverse order.
%C A181973 I found in the same family of prime-generating polynomials (with the discriminant equal to -163*2^p, where p is even), the polynomials 4n^2 - 152n + 1607, generating 40 primes in row starting from n = 0 (20 distinct ones) and 4n^2 - 140n + 1877, generating 36 primes in row starting from n = 0 (18 distinct ones).
%C A181973 The following 5 (10 with their "reversal" polynomials) are the only ones I know from the family of Euler's polynomial n^2 + n + 41 (having their discriminant equal to a multiple of -163) that generate more than 30 distinct primes in a row starting from n = 0 (beside the Escott's polynomial n^2 - 79n + 1601):
%C A181973 (1) 4n^2 - 154n + 1523 (4n^2 - 158n + 1601);
%C A181973 (2) 9n^2 - 231n + 1523 (9n^2 - 471n + 6203);
%C A181973 (3) 16n^2 - 292n + 1373 (16n^2 - 668n + 7013);
%C A181973 (4) 25n^2 - 365n + 1373 (25n^2 - 1185n + 14083);
%C A181973 (5) 16n^2 - 300n + 1447 (16n^2 - 628n + 6203).
%C A181973 Note: For the first 2 (4 with their reversals), already reported, see the link below to Carlos Rivera's site.
%H A181973 Bruno Berselli, <a href="/A181973/b181973.txt">Table of n, a(n) for n = 0..1000</a>
%H A181973 Marius Coman, <a href="https://www.researchgate.net/profile/Marius_Coman/publication/277912540">Ten prime-generating quadratic polynomials</a>, Preprint 2015.
%H A181973 Factor Database, <a href="http://www.factorization.ath.cx/index.php?query=16*n%5E2+-+300*n+%2B+1447&amp;use=n&amp;n=0&amp;VP=on&amp;VC=on&amp;EV=on&amp;OD=on&amp;PR=on&amp;FF=on&amp;PRP=on&amp;CF=on&amp;U=on&amp;C=on&amp;perpage=50&amp;format=1&amp;sent=Show">Factorizations of 16n^2-300n+1447</a>. [Broken link?]
%H A181973 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 A181973 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A181973 G.f.: (1447 - 3178*x + 1763*x^2)/(1-x)^3. - _Bruno Berselli_, Apr 06 2012
%F A181973 From _Elmo R. Oliveira_, Feb 09 2025: (Start)
%F A181973 E.g.f.: exp(x)*(1447 - 284*x + 16*x^2).
%F A181973 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A181973 Table[16*n^2 - 300*n + 1447, {n, 0, 50}] (* _T. D. Noe_, Apr 04 2012 *)
%o A181973 (Magma) [n^2-75*n+1447: n in [0..176 by 4]]; // _Bruno Berselli_, Apr 06 2012
%o A181973 (PARI) a(n)=16*n^2 - 300*n + 1447 \\ _Charles R Greathouse IV_, Dec 08 2014
%K A181973 nonn,easy
%O A181973 0,1
%A A181973 _Marius Coman_, Apr 04 2012
%E A181973 Offset changed from 1 to 0 by _Bruno Berselli_, Apr 06 2012