A181973 Prime-generating polynomial: a(n) = 16*n^2 - 300*n + 1447.

1447, 1163, 911, 691, 503, 347, 223, 131, 71, 43, 47, 83, 151, 251, 383, 547, 743, 971, 1231, 1523, 1847, 2203, 2591, 3011, 3463, 3947, 4463, 5011, 5591, 6203, 6847, 7523, 8231, 8971, 9743, 10547, 11383, 12251, 13151, 14083, 15047, 16043, 17071, 18131, 19223, 20347

Offset: 0

Author

Marius Coman, Apr 04 2012

Keywords

• nonn
• easy

Comments

This polynomial generates 30 primes in a row starting from n = 0.

The polynomial 16*n^2 - 628*n + 6203 generates the same primes in reverse order.

I found in the same family of prime-generating polynomials (with the discriminant equal to -163*2^p, where p is even), the polynomials 4n^2 - 152n + 1607, generating 40 primes in row starting from n = 0 (20 distinct ones) and 4n^2 - 140n + 1877, generating 36 primes in row starting from n = 0 (18 distinct ones).

The following 5 (10 with their "reversal" polynomials) are the only ones I know from the family of Euler's polynomial n^2 + n + 41 (having their discriminant equal to a multiple of -163) that generate more than 30 distinct primes in a row starting from n = 0 (beside the Escott's polynomial n^2 - 79n + 1601):

(1) 4n^2 - 154n + 1523 (4n^2 - 158n + 1601);

(2) 9n^2 - 231n + 1523 (9n^2 - 471n + 6203);

(3) 16n^2 - 292n + 1373 (16n^2 - 668n + 7013);

(4) 25n^2 - 365n + 1373 (25n^2 - 1185n + 14083);

(5) 16n^2 - 300n + 1447 (16n^2 - 628n + 6203).

Note: For the first 2 (4 with their reversals), already reported, see the link below to Carlos Rivera's site.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Marius Coman, Ten prime-generating quadratic polynomials, Preprint 2015.
• Factor Database, Factorizations of 16n^2-300n+1447. [Broken link?]
• Carlos Rivera, Puzzle 232: Primes and Cubic polynomials, The Prime Puzzles & Problems Connection.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Magma

[n^2-75*n+1447: n in [0..176 by 4]]; // Bruno Berselli, Apr 06 2012

Mathematica

Table[16*n^2 - 300*n + 1447, {n, 0, 50}] (* T. D. Noe, Apr 04 2012 *)

PARI

a(n)=16*n^2 - 300*n + 1447 \\ Charles R Greathouse IV, Dec 08 2014

Formula

G.f.: (1447 - 3178*x + 1763*x^2)/(1-x)^3. - Bruno Berselli, Apr 06 2012

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

E.g.f.: exp(x)*(1447 - 284*x + 16*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