A220746 -id:A220746 - cp's OEIS frontend

A221211 Numbers n such that n and n + 4 are prime and there is a power of two in the interval (n,n+4).

3, 7, 13, 127

Offset: 1

Brad Clardy, Feb 21 2013

nonn

It is a conjecture that this is a finite sequence. These may be the only known cousin primes with this property.

The Cf.s list similar sequences, of the form -- numbers n such that n and n+m are prime and contain a power of two in the interval (n,n+m). The case where m=2, the twin prime case -- not listed, has only one member n=3. Another member would have to be a twin where n+2 was a Fermat type prime and n a Mersenne prime.

Cf. A023200.

Cf. A220951 (gap of 6), A213210 (8), A220746 (10), A213677 (12), A222424 (14), A222227 (16), A222219 (18).

//Program finds primes separated by an even number (called gap) which
//have a power of two between them. The program starts with the smallest
//of two above gap. Primes less than this starting point can be checked by
//inspection. In this example 3 also works.
gap:=4;
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;

[n: n in PrimesUpTo(10^3) | IsPrime(n+4) and exists{t: t in [n+1..n+3 by 2] | IsOne(t/2^Valuation(t,2))}]; // Bruno Berselli, May 16 2013

A220951 Primes p such that p+6 is also prime and there is a power of two in the interval (p,p+6).

Original entry on oeis.org

5, 7, 11, 13, 31, 61, 251, 4093

Offset: 1

Brad Clardy, Feb 20 2013

A search for sexy primes bracketing a power of two was conducted up to 2^1500. It is conjectured that this is a finite sequence.

On the basis of existing work about primes of the form 2^n+k and 2^n-k, plus a few additional tests, we have a(9) > 2^750740. - Giovanni Resta, Feb 21 2013

Cf. A023201, A220746, A221211.

//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:=6;
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;

pptQ[n_]:=AllTrue[{n,n+6},PrimeQ]&&Count[Log[2,#]&/@Range[n,n+6], ?IntegerQ] > 0; Select[Range[4100],pptQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Nov 01 2015 *)

cp's OEIS Frontend

A221211 Numbers n such that n and n + 4 are prime and there is a power of two in the interval (n,n+4).

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Magma

Magma

A220951 Primes p such that p+6 is also prime and there is a power of two in the interval (p,p+6).

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Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica