A286319 Prime p such that p^2-p-1 or p^2+p-1 is the smallest prime of a twin prime pair.

2, 3, 5, 7, 41, 59, 67, 89, 101, 131, 139, 379, 457, 743, 761, 1193, 1201, 1381, 1549, 1567, 1747, 1789, 2137, 2411, 2557, 2647, 2663, 2729, 2731, 3011, 3221, 3251, 3449, 4057, 4159, 4447, 4561, 4751, 5179, 5641, 6173, 6397, 6599, 6833, 7229, 8669, 9059, 9157, 9323

Offset: 1

Author

Pierre CAMI, May 11 2017

Keywords

• nonn

Comments

Union of A088483 and A120364.

3 is the only prime such that p^2-p-1 and p^2+p-1 are both the smallest of a prime twin pair.

For prime p > 3 if p+1 is divisible by 6 then the smallest prime of the prime twin pair is p^2+p-1 and p^2-p-1 if not.

Examples

			2^2+2-1=5 and (5,7) is a twin prime pair so a(1)=2.
3^2-3-1=5, 3^2+3-1=11 and (5,7), (11,13) are twin prime pairs so a(2)=3.
5^2+5-1=29 and (29,31) is a twin prime pair so a(3)=5.
7^2-7-1=41 and (41,43) is a twin prime pair so a(4)=7.
		

Links

• Pierre CAMI, Table of n, a(n) for n = 1..50000

Crossrefs

Cf. A001359, A088483, A088485, A120364.

Programs

Mathematica

sptppQ[n_]:=AllTrue[{n^2-n-1,n^2-n+1},PrimeQ]||AllTrue[{n^2+n-1,n^2+ n+ 1},PrimeQ]; Select[Prime[Range[1200]],sptppQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 04 2019 *)