A181963 Prime-generating polynomial: a(n) = 25*n^2 - 1185*n + 14083.

14083, 12923, 11813, 10753, 9743, 8783, 7873, 7013, 6203, 5443, 4733, 4073, 3463, 2903, 2393, 1933, 1523, 1163, 853, 593, 383, 223, 113, 53, 43, 83, 173, 313, 503, 743, 1033, 1373, 1763, 2203, 2693, 3233, 3823, 4463, 5153, 5893, 6683, 7523, 8413, 9353, 10343, 11383, 12473

Offset: 0

Author

Marius Coman, Apr 04 2012

Keywords

• nonn
• easy

Comments

The polynomial generates 32 primes starting from n = 0.

The polynomial 25*n^2 - 365*n + 1373 generates the same primes in reverse order.

This family of prime-generating polynomials (with the discriminant equal to -4075 = -163*5^2) is interesting for generating primes of same form: the polynomial 25*n^2 - 395*n + 1601 generates 16 primes of the form 10*k + 1 (1601, 1231, 911, 641, 421, 251, 131, 61, 41, 71, 151, 281, 461, 691, 971, 1301) and the polynomial 25*n^2 + 25*n + 47 generates 16 primes of the form 10*k + 7 (47, 97, 197, 347, 547, 797, 1097, 1447, 1847, 2297, 2797, 3347, 3947, 4597, 5297, 6047).

Note: all the polynomials of the form 25*n^2 + 5*n + 41, 25*n^2 + 15*n + 43, ..., 25*n^2 + 5*(2*k+1)*n + p, ..., 25*n^2 + 5*79*n + 1601, where p is a (prime) term of the Euler polynomial p = k^2 + k + 41, from k=0 to k=39, have their discriminant equal to -4075 = -163*5^2.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Factor Database, Factorizations of 25n^2-1185n+14083.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Magma

[n^2-237*n+14083: n in [0..220 by 5]]; // Bruno Berselli, Apr 06 2012

Mathematica

Table[25*n^2 - 1185*n + 14083, {n, 0, 50}] (* T. D. Noe, Apr 04 2012 *)
LinearRecurrence[{3,-3,1},{14083,12923,11813},50] (* Harvey P. Dale, Aug 28 2022 *)

PARI

a(n)=25*n^2-1185*n+14083 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: (14083 - 29326*x + 15293*x^2)/(1-x)^3. - Bruno Berselli, Apr 06 2012

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

E.g.f.: exp(x)*(14083 - 1160*x + 25*x^2).

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

Extensions

Offset changed from 1 to 0 by Bruno Berselli, Apr 06 2012