A181963 - OEIS

A181963

Prime-generating polynomial: 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

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OFFSET

0,1

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 25n^2 - 395n + 1601 generates 16 primes of the form 10k+1 (1601, 1231, 911, 641, 421, 251, 131, 61, 41, 71, 151, 281, 461, 691, 971, 1301) and the polynomial 25n^2 + 25n + 47 generates 16 primes of the form 10k+7 (47, 97, 197, 347, 547, 797, 1097, 1447, 1847, 2297, 2797, 3347, 3947, 4597, 5297, 6047).

Note: all the polynomials of the form 25n^2 + 5n + 41, 25n^2 + 15n + 43, ..., 25n^2 + 5*(2k+1)*n + p, ..., 25n^2 + 5*79n + 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).

FORMULA

G.f.: (14083-29326*x+15293*x^2)/(1-x)^3. - 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 *)

PROG

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

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