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

Author

Roger L. Bagula, Apr 17 2006

Keywords

• sign
• easy
• less

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. See p. 147.

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), pages 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).

Crossrefs

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

Programs

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

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 *)

PARI

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

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

E.g.f.: exp(x)*(2753 - 774*x + 36*x^2). - Elmo R. Oliveira, Feb 09 2025

Extensions

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

Title extended by Hugo Pfoertner, Dec 13 2019