A211775 a(n) = 2*n^2 - 212*n + 5419.

5419, 5209, 5003, 4801, 4603, 4409, 4219, 4033, 3851, 3673, 3499, 3329, 3163, 3001, 2843, 2689, 2539, 2393, 2251, 2113, 1979, 1849, 1723, 1601, 1483, 1369, 1259, 1153, 1051, 953, 859, 769, 683, 601, 523, 449, 379, 313, 251, 193, 139, 89, 43, 1, -37, -71, -101, -127

Offset: 0

Author

Marius Coman, May 18 2012

Keywords

• sign
• easy

Comments

A "prime-generating" polynomial: This polynomial generates 92 primes (57 being distinct) for 0 <= n <= 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).

The substitution n = m + 53 converts this polynomial to the simpler form 2*m^2 - 199. - Charles Kusniec, Nov 11 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
• Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials.
• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Magma

[2*n^2-212*n+5419: n in [0..49]]; // Bruno Berselli, May 18 2012

Maple

A211775:=n->2*n^2 - 212*n + 5419: seq(A211775(n), n=0..100); # Wesley Ivan Hurt, Jan 20 2017

Mathematica

Table[2*n^2 - 212*n + 5419, {n, 0, 80}] (* Wesley Ivan Hurt, Aug 06 2017 *)

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

Formula

G.f.: (5419 - 11048*x + 5633*x^2)/(1-x)^3. - Bruno Berselli, May 18 2012

From Elmo R. Oliveira, Feb 09 2025: (Start)

E.g.f.: exp(x)*(5419 - 210*x + 2*x^2).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Extensions

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