A222219 Numbers n such that n and n + 18 are prime and there is a power of two in the interval (n,n+18).

5, 11, 13, 19, 23, 29, 53, 61, 113, 239, 251, 503, 1013, 1021, 4093, 8191, 65519, 65521, 262133, 1048571, 4194301, 302231454903657293676533

Offset: 1

Author

Brad Clardy, Feb 23 2013

Keywords

• nonn

Comments

It is a conjecture that this sequence is finite. A search around 2^n was done up to 2^1500.

Crossrefs

Cf. A153418, A221211.

Programs

Magma

//Program finds primes separated by an even number (called gap) which
//have a power of two between them. Program starts with the smallest
//power of two above gap. Primes less than this starting point can be
//checked by inspection.
gap:=18;
start:=Ilog2(gap)+1;
for i:= start to 1000 do
    powerof2:=2^i;
    for k:=powerof2-gap+1 to powerof2-1 by 2 do
        if (IsPrime(k) and IsPrime(k+gap)) then k;
        end if;
    end for;
end for;

Mathematica

Flatten[Table[Select[2^n-Range[17],AllTrue[{#,#+18},PrimeQ]&],{n,4,80}]]// Sort (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 04 2019 *)