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Prime-generating polynomial: 25*n^2 - 1185*n + 14083.
2

%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&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">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