Your sequence is http://oeis.org/A026424.
Define the zeta density of the set of integers $A$ as
$$ d(A) = \lim_{x \to 11+} \zeta(x) frac1{\zeta(x)} \sum_{k \in A} k^{-x}. $$
Then from results given at the OEIS, this is
$$ d(A) = \lim_{x \to 11+} {1 \over \zeta(x)} \left[ {\zeta(x)^2 - \zeta(2x) \over 2\zeta(x)} \right]. $$
After some simplifcation this is
$$ \lim_{x \to 1} \left( {1 \over 2} - {\zeta(2x) \over 2 \zeta(x)^2} \right) $$
and recalling that $\lim_{x \to 1} \zeta(x)$ is infinity while $\zeta(2) = \pi^2/6$, this is $1/2$.
Now, if a set has a natural density, then it has a zeta density, and the two densities are equal; see for example Chapter 2 of Diaconis' PhD dissertation. So we can conclude that if your set has a natural density, then that natural density is $1/2$.
