A370523 Numbers k > 2 such that all positive values of k - 2^(2^m) are prime, with integer m >= 0.

4, 7, 9, 15, 21, 33, 45, 63, 75, 105, 153, 183, 195, 243, 273, 285, 435, 525, 573, 603, 813, 825, 1065, 1233, 1305, 1623, 2145, 2595, 2715, 2805, 3375, 3465, 3933, 4023, 4245, 4275, 4653, 4803, 4935, 5655, 6303, 6705, 7563, 8865, 10095, 10503, 10863, 12165, 12243, 12825, 13713, 13725, 14013

Offset: 1

Author

Thomas Ordowski, Feb 22 2024

Keywords

• nonn

Comments

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

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

It seems that there are infinitely many such numbers.

Note that A039669 is finite and probably complete.

Examples

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

Crossrefs

Cf. A039669, A129613.

Programs

Mathematica

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 *)

Extensions

More terms from Amiram Eldar, Feb 22 2024