|
%I #22 Sep 08 2022 08:44:57
%S 11,13,17,23,31,41,53,67,83,101,167,193,251,283,317,353,431,563,661,
%T 823,881,941,1201,1493,1571,1733,2081,2267,2663,2767,3203,3433,3671,
%U 3793,3917,4567,4703,5413,5711,6173,6491,6653,6983,7151,7321
%N Primes of the form k^2 + k + 11.
%C From _Peter Bala_, Apr 15 2018: (Start)
%C 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.
%C 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.
%C 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)
%H Vincenzo Librandi, <a href="/A048059/b048059.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%t 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 *)
%t Select[Table[n^2+n+11, {n,0,600}], PrimeQ] (* _Vincenzo Librandi_, Dec 07 2011 *)
%o (Magma) [ a: n in [0..200] | IsPrime(a) where a is n^2+n+11 ]; // _Vincenzo Librandi_, Dec 07 2011
%Y Cf. A005846, A007635, A048058, A048097, A160548.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
|
