A237414 Primes p with p^2 - 2 and prime(p)^2 - 2 both prime.

2, 3, 43, 47, 107, 139, 191, 211, 223, 239, 293, 313, 337, 541, 743, 757, 863, 1013, 1153, 1231, 1619, 2113, 2137, 2287, 2297, 2423, 2543, 2729, 2749, 2897, 3079, 3089, 3313, 3863, 3947, 4241, 4271, 4583, 4649, 4993, 5581, 6571, 6637, 6911, 7547, 8629, 8849, 8867, 9049, 9661

Offset: 1

Author

Zhi-Wei Sun, Feb 07 2014

Keywords

• nonn

Comments

According to the conjecture in A237413, this sequence should have infinitely many terms.

Examples

			a(1) = 2 since 2^2 - 2 = 2 and prime(2)^2 - 2 = 3^2 - 2 = 7 are both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A049002, A062326, A230502, A237413.

Programs

Mathematica

p[n_]:=PrimeQ[n^2-2]
n=0;Do[If[p[Prime[k]]&&p[Prime[Prime[k]]],n=n+1;Print[n," ",Prime[k]]],{k,1,1000}]
Select[Prime[Range[1200]],AllTrue[{#^2-2,Prime[#]^2-2},PrimeQ]&] (* Harvey P. Dale, Apr 06 2022 *)