A154666 Numbers k such that k-1, k+1, and k^2-k-1 are primes.

4, 6, 12, 42, 60, 102, 150, 192, 282, 420, 432, 462, 570, 660, 810, 882, 1062, 1230, 1320, 1872, 1950, 2550, 2712, 2730, 3120, 3252, 4020, 4092, 6090, 8220, 8862, 8970, 9042, 9240, 9720, 10140, 10530, 10710, 11550, 11832, 11970, 12252, 13680, 13902

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

Keywords

• nonn

Comments

Note that k-1 and k+1 are twin primes and the third prime is (k-1)*(k+1)-k, the product of the twin primes minus their average.

Examples

			3*5 - 4 = 11.
5*7 - 6 = 29.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Programs

Magma

[k:k in [2..15000]| IsPrime(k-1) and IsPrime(k+1)and IsPrime(k^2-k-1)]; // Marius A. Burtea, Dec 21 2019

Mathematica

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],If[PrimeQ[(n-1)*(n+1)-n],AppendTo[lst,n]]],{n,8!}];lst
Mean/@Select[Select[Partition[Prime[Range[2000]],2,1],Last[#]-First[#]==2&], PrimeQ[Times@@#-Mean[#]]&] (* Harvey P. Dale, Mar 17 2011 *)