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Squarefree Hilbert numbers

Published on Sunday, 4th December 2016, 04:00 am; Solved by 171;
Difficulty rating: 70%

Problem 580

A Hilbert number is any positive integer of the form $4k+1$ for integer $k\geq 0$. 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 $9\times13$. 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 $10^{16}$?