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Numbers k > 2 such that all positive values of k - 2^(2^m) are prime, with integer m >= 0.
1

%I #29 Mar 19 2024 12:41:39

%S 4,7,9,15,21,33,45,63,75,105,153,183,195,243,273,285,435,525,573,603,

%T 813,825,1065,1233,1305,1623,2145,2595,2715,2805,3375,3465,3933,4023,

%U 4245,4275,4653,4803,4935,5655,6303,6705,7563,8865,10095,10503,10863,12165,12243,12825,13713,13725,14013

%N Numbers k > 2 such that all positive values of k - 2^(2^m) are prime, with integer m >= 0.

%C If k > 4 is a term of this sequence, then (k-2, k-4) is a twin prime pair.

%C So all terms k > 7 are divisible by 3, and k = 7 is the only prime here.

%C It seems that there are infinitely many such numbers.

%C Note that A039669 is finite and probably complete.

%e The number 15 is a term, since 15-2^(2^0) and 15-2^(2^1) are primes 13 and 11.

%t q[k_] := Module[{m = 0}, While[2^(2^m) < k && PrimeQ[k - 2^(2^m)], m++]; 2^(2^m) >= k]; Select[Range[4, 15000], q] (* _Amiram Eldar_, Feb 22 2024 *)

%Y Cf. A039669, A129613.

%K nonn

%O 1,1

%A _Thomas Ordowski_, Feb 22 2024

%E More terms from _Amiram Eldar_, Feb 22 2024