A210626 Values of the prime-generating polynomial 4*n^2 - 284*n + 3449.

3449, 3169, 2897, 2633, 2377, 2129, 1889, 1657, 1433, 1217, 1009, 809, 617, 433, 257, 89, -71, -223, -367, -503, -631, -751, -863, -967, -1063, -1151, -1231, -1303, -1367, -1423, -1471, -1511, -1543, -1567, -1583, -1591, -1591, -1583, -1567, -1543, -1511, -1471

Offset: 0

Author

Marius Coman, May 08 2012

Keywords

• sign
• easy

Comments

The polynomial successively generates 35 primes/negative values of primes starting at n = 0.

This polynomial generates 95 primes in absolute value (60 distinct ones) for n from 0 to 99, equaling the record held by Euler's polynomial for n = m - 35, which is m^2 - 69*m + 1231 (see the reference).

The nonprime terms (in absolute value) up to n = 99 are: 1591 = 37*43, 3737 = 37*101, 4033 = 37*109; 5633 = 43*131; 5977 = 43*139; 9017 = 71*127.

The polynomial 4*n^2 + 12*n - 1583 generates the same 35 primes in row starting from n = 0 in reverse order.

Note: in the same family of prime-generating polynomials (with the discriminant equal to 199*2^p, where p is odd) there are the polynomial 32*n^2 - 944*n + 6763 (with its "reversed polynomial" 32*m^2 - 976*m + 7243, for m=30-n), generating 31 primes in row, and the polynomial 4*n^2 - 428*n + 5081 (with 4*m^2 + 188*m - 4159, for m = 30 - n), generating 31 primes in row.

References

• Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials, Lecture Notes in Pure and Applied Mathematics (Vol. 220), Marcel Dekker Inc., 2001, pages 281-317.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematic, Prime-Generating Polynomial.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Mathematica

LinearRecurrence[{3,-3, 1},{3449,3169,2897},100] (* Vincenzo Librandi, Aug 01 2012 *)

PARI

Vec((3449-7178*x+3737*x^2)/(1-x)^3+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012

Formula

G.f.: (3449 - 7178*x + 3737*x^2)/(1-x)^3. - Bruno Berselli, Jun 07 2012

From Elmo R. Oliveira, Feb 09 2025: (Start)

E.g.f.: exp(x)*(3449 - 280*x + 4*x^2).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)