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A048059 Primes of the form k^2 + k + 11. 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A191031 A106890 A048058 * A106076 A006489 A032621

Adjacent sequences:  A048056 A048057 A048058 * A048060 A048061 A048062

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 19 19:30 EST 2020. Contains 332047 sequences. (Running on oeis4.)