A211773 Prime-generating polynomial: a(n) = 2*n^2 - 108*n + 1259.

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, 859, 953

Offset: 0

Author

Marius Coman, May 18 2012

Keywords

• sign
• easy

Comments

This polynomial generates 92 primes (66 distinct ones) for 0 <= n <= 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
• Marius Coman, Ten prime-generating quadratic polynomials, Preprint 2015.
• 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).

Crossrefs

Cf. A060566, A202018.

Programs

Magma

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

Mathematica

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

PARI

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

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

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

E.g.f.: exp(x)*(1259 - 106*x + 2*x^2).

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