A048059 Primes of the form k^2 + k + 11.

11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 167, 193, 251, 283, 317, 353, 431, 563, 661, 823, 881, 941, 1201, 1493, 1571, 1733, 2081, 2267, 2663, 2767, 3203, 3433, 3671, 3793, 3917, 4567, 4703, 5413, 5711, 6173, 6491, 6653, 6983, 7151, 7321, 8753, 8941, 9323, 10111

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Peter Bala, Apr 15 2018: (Start)

The polynomial P(n) := n^2 + n + 11 takes distinct prime values for the 10 consecutive integers n = 0 to 9. It follows that the polynomial P(n-10) = (n - 10)^2 + (n - 10) + 11 takes prime values for the 20 consecutive integers n = 0 to 19, consisting of the 10 primes above each taken twice. We note two consequences of this fact.

1) The polynomial P(2*n-10) = 4*n^2 - 38*n + 101 also takes prime values for the 10 consecutive integers n = 0 to 9.

2)The polynomial P(3*n-10) = 9*n^2 - 57*n + 101 takes prime values for the 7 consecutive integers n = 0 to 6 (= floor(19/3)). In addition, calculation shows that P(3*n-10) also takes prime values for n from -3 to -1. Equivalently put, the polynomial P(3*n-19) = 9*n^2 - 111*n + 353 takes prime values for the 10 consecutive integers n = 0 to 9. Cf. A007635 and A005846. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Crossrefs

Cf. A005846, A007635, A048058, A048097, A160548.

Programs

Magma

[ a: n in [0..200] | IsPrime(a) where a is n^2+n+11 ]; // Vincenzo Librandi, Dec 07 2011

Mathematica

lst={}; Do[p=n^2+n+11; If[PrimeQ[p], AppendTo[lst,p]], {n,0,5*5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Table[n^2+n+11, {n,0,600}], PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

Formula

a(n) = A048058(A048097(n)). - Elmo R. Oliveira, Apr 20 2025