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%I #11 Mar 19 2024 12:42:03
%S 7,9,15,27,57,63,195,267,363,405,483,603,1197,1233,1443,1737,2715,
%T 4257,5403,6117,21855,22287,26817,40755,63777,260007,617253,986733,
%U 1151655,1167837,1174503,1199373,1331595,3233307,4128873,4138707,4609527,5938107,7203945,7605213,8379405,8587545,9596223
%N 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.
%C It seems that there are infinitely many such numbers.
%C If k > 7 is such a number, then it is odd and divisible by 3.
%C 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).
%t 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 *)
%Y Cf. A039669, A129613, A370523.
%K nonn
%O 1,1
%A _Thomas Ordowski_, Mar 18 2024
%E More terms from _Amiram Eldar_, Mar 18 2024