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A213810 a(n) = 4*n^2 - 482*n + 14561. 1
14561, 14083, 13613, 13151, 12697, 12251, 11813, 11383, 10961, 10547, 10141, 9743, 9353, 8971, 8597, 8231, 7873, 7523, 7181, 6847, 6521, 6203, 5893, 5591, 5297, 5011, 4733, 4463, 4201, 3947, 3701, 3463, 3233, 3011, 2797, 2591, 2393, 2203, 2021, 1847, 1681, 1523 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

A "prime-generating" polynomial: This polynomial generates 88 distinct primes for n from 0 to 99, just two primes fewer than the record held by the polynomial discovered by N. Boston and M. L. Greenwood, that is 41*n^2 - 4641*n + 88007 (this polynomial is sometimes cited as 41*n^2 + 33*n - 43321, which is the same for the input values [-57, 42], see the references below).

The nonprime terms in the first 100 are: 10961 = 97*113; 10547 = 53*199; 9353 = 47*199; 7181 = 43*167; 6847 = 41*167; 5893 = 71*83; 3233 = 53*61; 2021 = 43*47; 1681 = 41^2; 1763 = 41*43; 2491 = 47*53; 4331 = 61*71.

For n = m + 41 we obtain the polynomial 4*m^2 - 154*m + 1523, which generates 40 primes in a row starting from m=0 (polynomial already reported, see the link below).

REFERENCES

W. Narkiewicz, The Development of Prime Number Theory: from Euclid to Hardy and Littlewood, Springer Monographs in Mathematics, 2000, page 43.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (1997), pp. 529-544.

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

a(n) = 4*n^2 - 482*n + 14561.

G.f.: (-15047*x^2+29600*x-14561)/(x-1)^3. - Alexander R. Povolotsky, Jun 21 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - G. C. Greubel, Feb 26 2017

MATHEMATICA

Table[4n^2-482n+14561, {n, 0, 41}] (* Harvey P. Dale, Sep 09 2014 *)

LinearRecurrence[{3, -3, 1}, {14561, 14083, 13613}, 50] (* or *) CoefficientList[Series[ (-15047*x^2+29600*x-14561)/(x-1)^3, {x, 0, 50}], x] (* G. C. Greubel, Feb 26 2017 *)

PROG

(PARI) x='x+O('x^50); Vec((-15047*x^2+29600*x-14561)/(x-1)^3) \\ G. C. Greubel, Feb 26 2017

CROSSREFS

Cf. A181973, A211773, A211775.

Sequence in context: A234119 A253116 A202489 * A247716 A252145 A249082

Adjacent sequences:  A213807 A213808 A213809 * A213811 A213812 A213813

KEYWORD

nonn,easy

AUTHOR

Marius Coman, Jun 20 2012

EXTENSIONS

Edited by N. J. A. Sloane, Nov 12 2016

STATUS

approved

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Last modified July 26 17:37 EDT 2021. Contains 346294 sequences. (Running on oeis4.)