OFFSET
0,1
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).
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)
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {3449, 3169, 2897}, 100] (* Vincenzo Librandi, Aug 01 2012 *)
PROG
(PARI) Vec((3449-7178*x+3737*x^2)/(1-x)^3+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012
CROSSREFS
KEYWORD
sign,easy,changed
AUTHOR
Marius Coman, May 08 2012
STATUS
approved