OFFSET
0,1
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 C. Rivera's site.
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..1000
M. 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).
FORMULA
G.f.: (1447-3178*x+1763*x^2)/(1-x)^3. - Bruno Berselli, Apr 06 2012
MATHEMATICA
Table[16*n^2 - 300*n + 1447, {n, 0, 50}] (* T. D. Noe, Apr 04 2012 *)
PROG
(Magma) [n^2-75*n+1447: n in [0..176 by 4]]; // Bruno Berselli, Apr 06 2012
(PARI) a(n)=16*n^2 - 300*n + 1447 \\ Charles R Greathouse IV, Dec 08 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Marius Coman, Apr 04 2012
EXTENSIONS
Offset changed from 1 to 0 by Bruno Berselli, Apr 06 2012
STATUS
approved