

A211773


Primegenerating 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
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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 099, customary for comparisons), just three primes less than the record held by Euler's polynomial for n = m35, 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*m34 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*m34 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, PrimeProducing Cubic Polynomials in Lecture Notes in Pure and Applied Mathematics (Vol. 220), Marcel Dekker Inc., 2001, pages 281317.


LINKS

Bruno Berselli, Table of n, a(n) for n = 0..1000
M. Coman, Ten primegenerating quadratic polynomials, Preprint 2015.
Joe L. Mott and Kermite Rose, PrimeProducing Cubic Polynomials
E. W. Weisstein, MathWorld: PrimeGenerating Polynomial
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

G.f.: (12592624*x+1369*x^2)/(1x)^3.  Bruno Berselli, May 18 2012
a(n37) = 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^2108*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



