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.

2753, 1979, 1277, 647, 89, -397, -811, -1153, -1423, -1621, -1747, -1801, -1783, -1693, -1531, -1297, -991, -613, -163, 359, 953, 1619, 2357, 3167, 4049, 5003, 6029, 7127, 8297, 9539, 10853, 12239, 13697, 15227, 16829, 18503, 20249, 22067, 23957, 25919, 27953, 30059, 32237, 34487, 36809, 39203, 41669

Offset

0,1

Comments

The absolute values of a(n) for 0 <= n <= 44 are primes, a(45) = 39203 = 197*199. The positive prime terms are in A050268.

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

References

Paulo Ribenboim, The Little Book of Bigger Primes, Second Edition, Springer-Verlag New York, 2004.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

François Dress and Michel Olivier, Polynômes prenant des valeurs premières, Experimental Mathematics, Volume 8, Issue 4 (1999), 319-338.

Carlos Rivera, Problem 12: Prime producing polynomials, The Prime Puzzles and Problems Connection.

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

G.f.: (2753-6280*x+3599*x^2)/(1-x)^3. [Colin Barker, May 10 2012]

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

Mathematica

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}]
CoefficientList[Series[(2753-6280*x+3599*x^2)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, May 12 2012 *)
Table[36n^2-810n+2753, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {2753, 1979, 1277}, 50] (* Harvey P. Dale, Jun 20 2013 *)

Prog

(PARI) {for(n=0, 46, print1(36*n^2-810*n+2753, ", "))}
(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

Crossrefs

Cf. A005846, A050267, A050268, A117081, A267252.

Sequence in context: A045151 A122107 A050268 * A164065 A014487 A260978

Adjacent sequences: A117078 A117079 A117080 * A117082 A117083 A117084

Keyword

sign,easy,less

Author

Roger L. Bagula, Apr 17 2006

Extensions

Edited by N. J. A. Sloane, Apr 27 2007

Title extended by Hugo Pfoertner, Dec 13 2019

Status

approved