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%I #42 Sep 08 2022 08:45:54
%S 14083,12923,11813,10753,9743,8783,7873,7013,6203,5443,4733,4073,3463,
%T 2903,2393,1933,1523,1163,853,593,383,223,113,53,43,83,173,313,503,
%U 743,1033,1373,1763,2203,2693,3233,3823,4463,5153,5893,6683,7523,8413,9353,10343
%N Prime-generating polynomial: 25*n^2 - 1185*n + 14083.
%C The polynomial generates 32 primes starting from n=0.
%C The polynomial 25*n^2 - 365*n + 1373 generates the same primes in reverse order.
%C This family of prime-generating polynomials (with the discriminant equal to -4075 = -163*5^2) is interesting for generating primes of same form: the polynomial 25n^2 - 395n + 1601 generates 16 primes of the form 10k+1 (1601, 1231, 911, 641, 421, 251, 131, 61, 41, 71, 151, 281, 461, 691, 971, 1301) and the polynomial 25n^2 + 25n + 47 generates 16 primes of the form 10k+7 (47, 97, 197, 347, 547, 797, 1097, 1447, 1847, 2297, 2797, 3347, 3947, 4597, 5297, 6047).
%C Note: all the polynomials of the form 25n^2 + 5n + 41, 25n^2 + 15n + 43, ..., 25n^2 + 5*(2k+1)*n + p, ..., 25n^2 + 5*79n + 1601, where p is a (prime) term of the Euler polynomial p = k^2 + k + 41, from k=0 to k=39, have their discriminant equal to -4075 = -163*5^2.
%H Bruno Berselli, <a href="/A181963/b181963.txt">Table of n, a(n) for n = 0..1000</a>
%H Factor Database, <a href="http://www.factorization.ath.cx/index.php?query=25*n%5E2-1185*n%2B14083&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=50&format=1">Factorizations of 25n^2-1185n+14083</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: (14083-29326*x+15293*x^2)/(1-x)^3. - _Bruno Berselli_, Apr 06 2012
%t Table[25*n^2 - 1185*n + 14083, {n, 0, 50}] (* _T. D. Noe_, Apr 04 2012 *)
%t LinearRecurrence[{3,-3,1},{14083,12923,11813},50] (* _Harvey P. Dale_, Aug 28 2022 *)
%o (Magma) [n^2-237*n+14083: n in [0..220 by 5]]; // _Bruno Berselli_, Apr 06 2012
%o (PARI) a(n)=25*n^2-1185*n+14083 \\ _Charles R Greathouse IV_, Jun 17 2017
%K nonn,easy
%O 0,1
%A _Marius Coman_, Apr 04 2012
%E Offset changed from 1 to 0 by _Bruno Berselli_, Apr 06 2012