

A182409


Primegenerating polynomial: 4n^2 + 12n  1583.


1



1583, 1567, 1543, 1511, 1471, 1423, 1367, 1303, 1231, 1151, 1063, 967, 863, 751, 631, 503, 367, 223, 71, 89, 257, 433, 617, 809, 1009, 1217, 1433, 1657, 1889, 2129, 2377, 2633, 2897, 3169, 3449, 3737, 4033, 4337, 4649, 4969, 5297, 5633, 5977
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

The polynomial generates 35 primes/negative values of primes in row starting from n=0.
The polynomial 4*n^2  284*n + 3449 generates the same primes in reverse order.
Other related polynomials:
For n = 6n+6 than n = n11 we get 144n^2  2808n + 12097 which generates 16 primes in row starting from n=0 (with the discriminant equal to 2^9*3^2*199);
For n = 12n+12 than n = n15 we get 576n^2  15984n + 109297 which generates 17 primes in row starting from n=0 (with the discriminant equal to 2^11*3^2*199).
So this polynomial opens at least two directions of study:
(1) polynomials of type 4n^2 + 12n  p, where p is prime (could be of the form 30k+23);
(2) polynomials with the discriminant equal to 2^n*3^m*199, where n is odd and m is even (an example of such a polynomial, with the discriminant equal to 2^5*3^4*199, is 36n^2  1020n + 3643 which generates 32 primes for values of n from 0 to 34).


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. W. Weisstein, MathWorld: PrimeGenerating Polynomial
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

From Chai Wah Wu, May 28 2016: (Start)
a(n) = 3*a(n1)  3*a(n2) + a(n3).
G.f.: (1591*x^2  3182*x + 1583)/(x  1)^3.
(End)


MATHEMATICA

Table[4 n^2 + 12 n  1583, {n, 0, 50}] (* Vincenzo Librandi, May 29 2016 *)


PROG

(PARI) a(n)=4*n^2+12*n1583 \\ Charles R Greathouse IV, Oct 01 2012
(MAGMA) [4*n^2+12*n1583: n in [0..50]]; // Vincenzo Librandi, May 29 2016


CROSSREFS

Sequence in context: A253421 A023082 A216064 * A350027 A224944 A222164
Adjacent sequences: A182406 A182407 A182408 * A182410 A182411 A182412


KEYWORD

sign,easy


AUTHOR

Marius Coman, May 09 2012


STATUS

approved



