A371303 Numbers k > 4 such that both k - 2^(2^m) and k + 2^(2^m) are prime for every natural m > 0 with 2^(2^m) < k.

7, 9, 15, 27, 57, 63, 195, 267, 363, 405, 483, 603, 1197, 1233, 1443, 1737, 2715, 4257, 5403, 6117, 21855, 22287, 26817, 40755, 63777, 260007, 617253, 986733, 1151655, 1167837, 1174503, 1199373, 1331595, 3233307, 4128873, 4138707, 4609527, 5938107, 7203945, 7605213, 8379405, 8587545, 9596223

Offset: 1

Author

Thomas Ordowski, Mar 18 2024

Keywords

• nonn

Comments

It seems that there are infinitely many such numbers.

If k > 7 is such a number, then it is odd and divisible by 3.

Conjecture: numbers k > 2 such that both k - 2^(2^m) and k + 2^(2^m) are prime for every integer m >= 0 with 2^(2^m) < k are only 9, 15, and 195 (Amiram Eldar checked that there are no more terms k < 10^8).

Crossrefs

Cf. A039669, A129613, A370523.

Programs

Mathematica

q[k_] := Module[{m = 1}, While[2^(2^m) < k && PrimeQ[k - 2^(2^m)] && PrimeQ[k + 2^(2^m)], m++]; 2^(2^m) > k]; Select[Range[5, 10^6, 2], q] (* Amiram Eldar, Mar 18 2024 *)

Extensions

More terms from Amiram Eldar, Mar 18 2024