A188903 a(n) is the least power of 2 such that 2n+1 - a(n) is prime, or 0 if no such prime exists.

0, 1, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 16, 2, 2, 4, 8, 2, 4, 2, 2, 4, 2, 4, 16, 2, 4, 16, 2, 2, 4, 8, 2, 4, 2, 2, 4, 8, 2, 4, 2, 4, 16, 2, 4, 16, 8, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 16, 8, 16, 16, 0, 2, 4, 2, 4, 64, 2, 2, 4, 8, 8, 0, 2, 2, 4, 8, 2, 4, 32, 2, 4, 2, 4, 16, 2, 4, 16, 2, 2, 4, 8, 8, 64, 2, 2, 4, 2, 2, 4, 8, 8, 16, 32, 2, 4, 128, 8, 64, 32, 2, 4, 2, 2, 4, 2, 4, 16, 2, 2, 4, 8, 8, 0, 2

Offset: 0

Author

Michel Lagneau, Apr 13 2011

Keywords

• nonn

Comments

The second Polignac's Conjecture states that every odd positive integer is the sum of a prime and a power of two. This conjecture was proved false, and the smallest counterexample is 127 because subtracting powers of 2 from 127 produces the composite numbers 126, 123, 119, 111, 95, and 63.

The sequence A006285 gives the odd numbers for which the conjecture fails. Hence, a(n) = 0 for n = (A006285(k)-1)/2 = {0, 63, 74, 125, 165, 168, 186, ...}.

Examples

			a(1) = 1 because 2*1 + 1 = 3 = 1 + 2 ;
a(2) = 2 because 2*2 + 1 = 5 = 2 + 3 ;
a(3) = 2 because 2*3 + 1 = 7 = 2 + 5 ;
a(63) = 0 ; a(74) = 0 ; a(125) = 0, ....
		

References

• David Wells, Prime Numbers: The Most Mysterious Figures In Math, John Wiley & Sons, 2005, p. 175-176.

Links

• Carlos Rivera, Puzzle 219, Polignac numbers, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A065381 (primes not of the form p + 2^k, p prime and k >= 0), A156695.

Programs

Maple

with(numtheory):for n from 1 to 126 do:x:=2*n+1:id:=0:for k from 0 to 50 while(id=0)
  do: for q from 1 to 100 while(id=0) do: p:=ithprime(q): y:=2^k+p:if y=x then
  id:=1:printf(`%d, `,2^k):else fi:od:od:if id=0 then printf(`%d, `,0):else fi:od:

Mathematica

Table[d = 2*n + 1; k = 1; While[k < d && ! PrimeQ[d - k], k = 2*k]; If[k < d, k, 0], {n, 0, 126}]

Sage

def A188903(n):
    return next((2**k for k in (0..floor(log(2*n+1,2))) if is_prime(2*n+1-2**k)), 0)
# D. S. McNeil, Apr 14 2011