A182409 Prime-generating polynomial: a(n) = 4*n^2 + 12*n - 1583.

-1583, -1567, -1543, -1511, -1471, -1423, -1367, -1303, -1231, -1151, -1063, -967, -863, -751, -631, -503, -367, -223, -71, 89, 257, 433, 617, 809, 1009, 1217, 1433, 1657, 1889, 2129, 2377, 2633, 2897, 3169, 3449, 3737, 4033, 4337, 4649, 4969, 5297, 5633, 5977

Offset: 0

Author

Marius Coman, May 09 2012

Keywords

• sign
• easy

Comments

The polynomial generates 35 primes/negative values of primes in row starting from n = 0.

The polynomial 4*n^2 - 284*n + 3449 generates the same primes in reverse order.

Other related polynomials:

For n = 6n + 6 than n = n-11 we get 144n^2 - 2808n + 12097 which generates 16 primes in row starting from n = 0 (with the discriminant equal to 2^9*3^2*199);

For n = 12n + 12 than n = n - 15 we get 576n^2 - 15984n + 109297 which generates 17 primes in row starting from n = 0 (with the discriminant equal to 2^11*3^2*199).

So this polynomial opens at least two directions of study:

(1) polynomials of type 4n^2 + 12n - p, where p is prime (could be of the form 30k + 23);

(2) polynomials with the discriminant equal to 2^n*3^m*199, where n is odd and m is even (an example of such a polynomial, with the discriminant equal to 2^5*3^4*199, is 36n^2 - 1020n + 3643 which generates 32 primes for values of n from 0 to 34).

Links

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

Programs

Magma

[4*n^2+12*n-1583: n in [0..50]]; // Vincenzo Librandi, May 29 2016

Mathematica

Table[4 n^2 + 12 n - 1583, {n, 0, 50}] (* Vincenzo Librandi, May 29 2016 *)

PARI

a(n)=4*n^2+12*n-1583 \\ Charles R Greathouse IV, Oct 01 2012

Formula

From Chai Wah Wu, May 28 2016: (Start)

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

G.f.: (1591*x^2 - 3182*x + 1583)/(x - 1)^3. (End)

E.g.f.: exp(x)*(-1583 + 16*x + 4*x^2). - Elmo R. Oliveira, Feb 09 2025