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

0,3

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, ...}.

References

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

Links

Table of n, a(n) for n=0..126.

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

Example

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, ....

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}]

Prog

(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

Crossrefs

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

Sequence in context: A367626 A049716 A361289 * A366534 A066671 A159802

Adjacent sequences: A188900 A188901 A188902 * A188904 A188905 A188906

Keyword

nonn

Author

Michel Lagneau, Apr 13 2011

Status

approved