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A211773 Prime-generating polynomial: 2*n^2 - 108*n + 1259. 3
1259, 1153, 1051, 953, 859, 769, 683, 601, 523, 449, 379, 313, 251, 193, 139, 89, 43, 1, -37, -71, -101, -127, -149, -167, -181, -191, -197, -199, -197, -191, -181, -167, -149, -127, -101, -71, -37, 1, 43, 89, 139, 193, 251, 313, 379, 449, 523, 601, 683, 769 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This polynomial generates 92 primes (66 distinct ones) for n from 0 to 99 (in fact the next two terms are still primes but we keep the range 0-99, customary for comparisons), just three primes less than the record held by Euler's polynomial for n = m-35, which is m^2 - 69*m + 1231 (see the link below), but having six distinct primes more than this one.

The nonprime terms in the first 100 are: 1 (taken twice), 1369 = 37^2, 1849 = 43^2, 4033 = 37*109, 5633 = 43*131, 7739 = 71*109 and 8251 = 37*223.

For n = 2*m-34 we obtain the polynomial 8*m^2 - 488*m + 7243, which generates 31 primes in a row starting from m=0 (polynomial already reported, see the link below).

For n = 4*m-34 we obtain the polynomial 32*m^2 - 976*m + 7243, which generates 31 primes in row starting from m=0.

The polynomial 2*n^2 + 40*n + 1, which generates the positive terms of this sequence in ascending order (i.e., a(37), ...), yields 10774009 distinct primes for 0 <= n < 49999999 while Euler's polynomial (n^2 - n + 41) gives 9967520 primes in same range. - Mikk Heidemaa, Feb 23 2016

REFERENCES

Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials in Lecture Notes in Pure and Applied Mathematics (Vol. 220), Marcel Dekker Inc., 2001, pages 281-317.

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..1000

M. Coman, Ten prime-generating quadratic polynomials, Preprint 2015.

Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials

E. W. Weisstein, MathWorld: Prime-Generating Polynomial

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: (1259-2624*x+1369*x^2)/(1-x)^3. - Bruno Berselli, May 18 2012

a(n-37) = 2*n^2 + 40*n + 1. - Mikk Heidemaa, Feb 18 2016

MATHEMATICA

Table[2 n^2 + 40 n + 1, {n, -37, 962}] (* Mikk Heidemaa, Feb 18 2016 *)

PROG

(MAGMA) [2*n^2-108*n+1259: n in [0..49]]; // Bruno Berselli, May 18 2012

(PARI) a(n)=2*n^2 - 108*n + 1259 \\ Charles R Greathouse IV, Jun 29 2017

CROSSREFS

Sequence in context: A159726 A048130 A256652 * A215991 A288921 A099592

Adjacent sequences:  A211770 A211771 A211772 * A211774 A211775 A211776

KEYWORD

sign,easy

AUTHOR

Marius Coman, May 18 2012

STATUS

approved

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Last modified January 17 14:12 EST 2019. Contains 319225 sequences. (Running on oeis4.)