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A117081 a(n) = 36*n^2 - 810*n + 2753, producing the conjectured record number of 45 primes in a contiguous range of n for quadratic polynomials, i.e., abs(a(n)) is prime for 0 <= n < 44.

%I A117081 #42 Apr 17 2025 12:42:40
%S A117081 2753,1979,1277,647,89,-397,-811,-1153,-1423,-1621,-1747,-1801,-1783,
%T A117081 -1693,-1531,-1297,-991,-613,-163,359,953,1619,2357,3167,4049,5003,
%U A117081 6029,7127,8297,9539,10853,12239,13697,15227,16829,18503,20249,22067,23957,25919,27953,30059,32237,34487,36809,39203,41669
%N A117081 a(n) = 36*n^2 - 810*n + 2753, producing the conjectured record number of 45 primes in a contiguous range of n for quadratic polynomials, i.e., abs(a(n)) is prime for 0 <= n < 44.
%C A117081 The absolute values of a(n) for 0 <= n <= 44 are primes, a(45) = 39203 = 197*199. The positive prime terms are in A050268.
%C A117081 The polynomial is a transformed version of the polynomial P(x) = 36*x^2 + 18*x - 1801 whose absolute value gives 45 distinct primes for -33 <= x <= 11, found by Ruby in 1989. It is one of the 3 known quadratic polynomials whose absolute value produces more than 40 primes in a contiguous range from 0 to n. For the other two polynomials, which produce 43 primes, see A050267 and A267252. - _Hugo Pfoertner_, Dec 13 2019
%D A117081 Paulo Ribenboim, The Little Book of Bigger Primes, Second Edition, Springer-Verlag New York, 2004. See p. 147.
%H A117081 Vincenzo Librandi, <a href="/A117081/b117081.txt">Table of n, a(n) for n = 0..1000</a>
%H A117081 François Dress and Michel Olivier, <a href="https://projecteuclid.org/euclid.em/1047262355">Polynômes prenant des valeurs premières</a>, Experimental Mathematics, Volume 8, Issue 4 (1999), pages 319-338.
%H A117081 Carlos Rivera, <a href="http://www.primepuzzles.net/problems/prob_012.htm">Problem 12: Prime producing polynomials</a>, The Prime Puzzles and Problems Connection.
%H A117081 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>.
%H A117081 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A117081 G.f.: (2753 - 6280*x + 3599*x^2)/(1-x)^3. - _Colin Barker_, May 10 2012
%F A117081 a(0)=2753, a(1)=1979, a(2)=1277, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Jun 20 2013
%F A117081 E.g.f.: exp(x)*(2753 - 774*x + 36*x^2). - _Elmo R. Oliveira_, Feb 09 2025
%t A117081 f[n_] := If[Mod[n, 2] == 1, 36*n^2 - 810*n + 2753, 36*n^2 - 810*n + 2753] a = Table[f[n], {n, 0, 100}]
%t A117081 CoefficientList[Series[(2753-6280*x+3599*x^2)/(1-x)^3,{x,0,50}],x] (* _Vincenzo Librandi_, May 12 2012 *)
%t A117081 Table[36n^2-810n+2753,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{2753,1979,1277},50] (* _Harvey P. Dale_, Jun 20 2013 *)
%o A117081 (PARI) {for(n=0, 46, print1(36*n^2-810*n+2753, ","))}
%o A117081 (Magma) I:=[2753, 1979, 1277]; [n le 3 select I[n] else 3*Self(n-1)-3 *Self(n-2)+Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, May 12 2012
%Y A117081 Cf. A005846, A050267, A050268, A117081, A267252.
%K A117081 sign,easy,less
%O A117081 0,1
%A A117081 _Roger L. Bagula_, Apr 17 2006
%E A117081 Edited by _N. J. A. Sloane_, Apr 27 2007
%E A117081 Title extended by _Hugo Pfoertner_, Dec 13 2019