

A188903


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


3



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
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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. 175176.


LINKS



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)
return next((2**k for k in (0..floor(log(2*n+1, 2))) if is_prime(2*n+12**k)), 0)


CROSSREFS

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


KEYWORD

nonn


AUTHOR



STATUS

approved



