A Hilbert number is any positive integer of the form $4k+1$ for integer $k\ge 0$. We shall define a squarefree Hilbert number as a Hilbert number which is not divisible by the square of any Hilbert number other than one. For example, $117$ is a squarefree Hilbert number, equaling $9\times 13$. However $6237$ is a Hilbert number that is not squarefree in this sense, as it is divisible by $9^2$. The number $3969$ is also not squarefree, as it is divisible by both $9^2$ and ${21}^2$.

There are $2327192$ squarefree Hilbert numbers below ${10}^7$.
How many squarefree Hilbert numbers are there below ${10}^{16}$?