This question is a follow-up to my previous question.
For what square-free values of $d$ is the following statement true?
For all $n\geq 1$, the equation $x^2+dy^2=n$ has a rational solution if and only if it has an integral solution.
This is true for $d=1$. Numerical calculations seem to indicate that this is true for $d=1,2,3,5,6,10,13,21,\ldots$ and false for $d=7,11,14,15,17,19,23,26,29,\ldots$