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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..4.

Prog

(Magma)
//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;
(Magma) [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

Crossrefs

Cf. A023200.

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

Sequence in context: A038691 A237890 A082718 * A322301 A154821 A353008

Adjacent sequences: A221208 A221209 A221210 * A221212 A221213 A221214

Keyword

nonn

Author

Brad Clardy, Feb 21 2013

Status

approved