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

1,1

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.

Links

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

Example

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.

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 *)

Crossrefs

Cf. A001359, A088483, A088485, A120364.

Sequence in context: A235395 A090714 A048400 * A090716 A083820 A244556

Adjacent sequences: A286316 A286317 A286318 * A286320 A286321 A286322

Keyword

nonn

Author

Pierre CAMI, May 11 2017

Status

approved