

A210626


Values of the primegenerating polynomial 4*n^2  284*n + 3449.


2



3449, 3169, 2897, 2633, 2377, 2129, 1889, 1657, 1433, 1217, 1009, 809, 617, 433, 257, 89, 71, 223, 367, 503, 631, 751, 863, 967, 1063, 1151, 1231, 1303, 1367, 1423, 1471, 1511, 1543, 1567, 1583, 1591, 1591, 1583, 1567, 1543, 1511, 1471
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OFFSET

0,1


COMMENTS

The polynomial successively generates 35 primes/negative values of primes starting at n=0.
This polynomial generates 95 primes in absolute value (60 distinct ones) for n from 0 to 99, equaling the record held by Euler's polynomial for n = m35, which is m^2  69*m + 1231 (see the reference).
The nonprime terms (in absolute value) up to n=99 are: 1591 = 37*43, 3737 = 37*101, 4033 = 37*109; 5633 = 43*131; 5977 = 43*139; 9017 = 71*127.
The polynomial 4*n^2 + 12*n  1583 generates the same 35 primes in row starting from n=0 in reverse order.
Note: in the same family of primegenerating polynomials (with the discriminant equal to 199*2^p, where p is odd) there are the polynomial 32*n^2  944*n + 6763 (with its "reversed polynomial" 32*m^2  976*m + 7243, for m=30n), generating 31 primes in row, and the polynomial 4*n^2  428*n +5081 (with 4*m^2 + 188*m  4159, for m=30n), generating 31 primes in row.


REFERENCES

Joe L. Mott and Kermite Rose, PrimeProducing Cubic Polynomials, Lecture Notes in Pure and Applied Mathematics (Vol. 220), Marcel Dekker Inc., 2001, pages 281317.


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

G.f.: (34497178*x+3737*x^2)/(1x)^3. [Bruno Berselli, Jun 07 2012]


MATHEMATICA

LinearRecurrence[{3, 3, 1}, {3449, 3169, 2897}, 100] (* Vincenzo Librandi, Aug 01 2012 *)


PROG

(PARI) Vec((34497178*x+3737*x^2)/(1x)^3+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012


CROSSREFS

Sequence in context: A260500 A202644 A218830 * A278270 A179705 A234893
Adjacent sequences: A210623 A210624 A210625 * A210627 A210628 A210629


KEYWORD

sign,easy


AUTHOR

Marius Coman, May 08 2012


STATUS

approved



