A181969 Prime-generating polynomial: a(n) = 16*n^2 - 292*n + 1373.

1373, 1097, 853, 641, 461, 313, 197, 113, 61, 41, 53, 97, 173, 281, 421, 593, 797, 1033, 1301, 1601, 1933, 2297, 2693, 3121, 3581, 4073, 4597, 5153, 5741, 6361, 7013, 7697, 8413, 9161, 9941, 10753, 11597, 12473, 13381, 14321, 15293, 16297, 17333, 18401, 19501, 20633

Offset: 0

Author

Marius Coman, Apr 04 2012

Keywords

• nonn
• easy

Comments

The polynomial generates 31 primes in row starting from n = 0.

The polynomial 16*n^2 - 668*n + 7013 generates the same primes in reverse order.

Note: all the polynomials of the form p^2*n^2 +- p*n + 41, p^2*n^2 +- 3*p*n + 43, p^2*n^2 +- 5*p*n + 47, ..., p^2*n^2 +- (2k+1)*p*n + q, ..., p^2*n^2 +- 79*p*n + 1601, where q is a (prime) term of the Euler polynomial q = k^2 + k + 41, from k=0 to k=39, have their discriminant equal to -163*p^2; the demonstration is easy: the discriminant is equal to b^2 - 4ac = (2k+1)^2*p^2 - 4*q*p^2 = - p^2 ((2k+1)^2 - 4q) = - p^2*(4k^2 + 4k + 1 - 4k^2 - 4k - 164) = -163*p^2.

Observation: many of the polynomials formed this way have the capacity to generate many primes in row. Examples:

9n^2 + 3n + 41 generates 27 primes in row starting from n = 0 (and 40 primes for n = n - 13);

9n^2 - 237n + 1601 generates 27 primes in row starting from n = 0;

16n^2 + 4n + 41 generates, for n = n - 21 (that is 16*n^2 - 668*n + 7013) 31 primes in row.

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^2-292n+1373.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Magma

[n^2-73*n+1373: n in [0..172 by 4]]; // Bruno Berselli, Apr 06 2012

Mathematica

Table[16*n^2 - 292*n + 1373, {n, 0, 50}] (* T. D. Noe, Apr 04 2012 *)

PARI

a(n)=16*n^2-292*n+1373 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: (1373 - 3022*x + 1681*x^2)/(1-x)^3. - Bruno Berselli, Apr 06 2012

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

E.g.f.: exp(x)*(1373 - 276*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