OFFSET
0,1
COMMENTS
A "prime-generating" polynomial: This polynomial generates 92 primes (57 being distinct) for n from 0 to 99 (in fact the next seven terms are still primes but we keep the range 0-99, customary for comparisons), just three primes fewer than the record held by Euler's polynomial for n = m-35, which is m^2 - 69*m + 1231 (see the link below).
The nonprime terms in the first 100 are 1, 1369 = 37^2, 1849 = 43^2, 4033 = 37*109 (all taken twice).
Setting n = 2*m+54 we obtain the polynomial 8*m^2 + 8*m - 197, which generates 31 primes in a row starting from m = 0 (the polynomial 8*m^2 - 488*m + 7243 generates the same 31 primes, but in reverse order).
From Charles Kusniec, Nov 11 2016: The substitution n = m+53 converts this polynomial to the simpler form 2*m^2-199.
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
Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials
Eric Weisstein's World of Mathematics, MathWorld: Prime-Generating Polynomial
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: (5419-11048*x+5633*x^2)/(1-x)^3. - Bruno Berselli, May 18 2012
MAPLE
MATHEMATICA
Table[2*n^2 - 212*n + 5419, {n, 0, 80}] (* Wesley Ivan Hurt, Aug 06 2017 *)
PROG
(Magma) [2*n^2-212*n+5419: n in [0..49]]; // Bruno Berselli, May 18 2012
(PARI) Vec((5419-11048*x+5633*x^2)/(1-x)^3+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012
(PARI) a(n) = 2*n^2 - 212*n + 5419 \\ Charles R Greathouse IV, Dec 19 2016
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Marius Coman, May 18 2012
EXTENSIONS
Edited by N. J. A. Sloane, Nov 12 2016
STATUS
approved