A221211 -id:A221211 - cp's OEIS frontend

A220746 Numbers n such that n and n + 10 are prime and there is a power of two in the interval (n, n+10).

3, 7, 13, 31, 61, 127, 1021, 1048573, 23945242826029513411849172299223580994042798784118781

Offset: 1

Brad Clardy, Feb 20 2013

nonn

//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. In this example 3 also works.
gap:=10;
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;

Union[Flatten[Table[Select[Range[2^n - 9, 2^n - 1, 2], PrimeQ[#] && PrimeQ[# + 10] &], {n, 3, 200}]]] (* T. D. Noe, Feb 20 2013 *)
Union[Flatten[Table[Select[Thread[{Range[2^n-10,2^n],Range[ 2^n,2^n+10]}],AllTrue[ #,PrimeQ]&],{n,3,1000}],1][[;;,1]]] (* Harvey P. Dale, Feb 19 2023 *)

print1(3); for(n=4,500,forprime(p=2^n-9,2^n-1,if(isprime(p+10), print1(", "p)))) \\ Charles R Greathouse IV, Feb 20 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 *)

A222227 Numbers n such that n and n + 16 are prime and there is a power of two in the interval (n,n+16).

Original entry on oeis.org

3, 7, 13, 31, 241, 65521, 1048573, 2305843009213693951

Offset: 1

Brad Clardy, Feb 23 2013

nonn

It is a conjecture that this is a finite sequence. A search was conducted out to 2^1500.

Cf. A049488, 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:=16;
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;

A213210 Numbers n such that n and n + 8 are prime and there is a power of two in the interval (n,n+8).

Original entry on oeis.org

3, 5, 11, 29, 59, 4091, 262139

Offset: 1

Brad Clardy, Mar 02 2013

nonn

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

Cf. A023202, A221211.

//Program finds primes separated by an even number (called gap) which
//have a power of two between them. The program starts with the smallest
//power of two above gap. Primes less than this starting point can be
//checked inspection. In this example 3 and 5 also work.
gap:=8;
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(10^6) | IsPrime(n+8) and exists{t: t in [n+1..n+7 by 2] | IsOne(t/2^Valuation(t,2))}]; // Bruno Berselli, May 16 2013

A213677 Numbers n such that n and n + 12 are prime and there is a power of two in the interval (n, n+12).

Original entry on oeis.org

5, 7, 11, 29, 31, 59, 61, 127, 251, 509, 1019, 1021, 262139, 1048571, 2147483647

Offset: 1

Brad Clardy, Mar 04 2013

nonn

It is a conjecture that this is a finite sequence. A search was conducted out to 2^1500.

A046133, 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:=12;
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;

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

Original entry on oeis.org

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

Offset: 1

Brad Clardy, Feb 23 2013

nonn

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

Cf. A153418, 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:=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;

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 *)

A222424 Numbers n such that n and n + 14 are prime and there is a power of two in the interval (n,n+14).

Original entry on oeis.org

3, 5, 23, 29, 53, 59, 509, 1019, 2039, 262133, 262139, 536870909, 73786976294838206459, 2787593149816327892691964784081045188247543

Offset: 1

Brad Clardy, Feb 24 2013

nonn

It is a conjecture that this is a finite sequence. A search was conducted out to 2^1500.

Cf. A153417, 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:=14;
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;

cp's OEIS Frontend

A220746 Numbers n such that n and n + 10 are prime and there is a power of two in the interval (n, n+10).

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Magma

Mathematica

PARI

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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Magma

Mathematica

A222227 Numbers n such that n and n + 16 are prime and there is a power of two in the interval (n,n+16).

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Magma

A213210 Numbers n such that n and n + 8 are prime and there is a power of two in the interval (n,n+8).

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Magma

Magma

A213677 Numbers n such that n and n + 12 are prime and there is a power of two in the interval (n, n+12).

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Programs

Magma

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

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Comments

Crossrefs

Programs

Magma

Mathematica

A222424 Numbers n such that n and n + 14 are prime and there is a power of two in the interval (n,n+14).

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Author

Keywords

Comments

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Programs

Magma