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81n^2 - 2247n + 15383.
0

%I #19 Jan 17 2019 11:21:04

%S 15383,13217,11213,9371,7691,6173,4817,3623,2591,1721,1013,467,83,

%T -139,-199,-97,167,593,1181,1931,2843,3917,5153,6551,8111,9833,11717,

%U 13763,15971,18341,20873,23567,26423,29441,32621,35963,39467,43133,46961,50951,55103

%N 81n^2 - 2247n + 15383.

%C The polynomial generates 33 primes/negative values of primes in a row starting from n=0.

%C The polynomial 81*n^2 - 2937*n + 26423 generates the same primes in reverse order.

%C Note: we found in the same family of prime-generating polynomials (with the discriminant equal to 64917 = 3^2*7213) the polynomial 27n^2 - 753n + 4649 (with its "reversed polynomial" 27n^2 - 921n + 7253), generating 32 primes in a row and the polynomial 27n^2 - 741n + 4483 (27n^2 - 1095n + 10501), generating 35 primes in a row, if we consider that 1 is prime (which seems to be constructive in the study of prime-generating polynomials, at least).

%C Note: the polynomial 36n^2 - 810n + 2753, which is the known quadratic polynomial that generates the most distinct primes in a row (45), has the discriminant equal to 259668 = 2^2*3^2*7213.

%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).

%t Table[81n^2-2247n+15383,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{15383,13217,11213},50] (* _Harvey P. Dale_, Jan 17 2019 *)

%o (PARI) a(n)=81*n^2-2247*n+15383 \\ _Charles R Greathouse IV_, Oct 01 2012

%K sign,easy

%O 0,1

%A _Marius Coman_, Apr 21 2012