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Problem 580

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Squarefree Hilbert numbers

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 ${10}^{16}$?

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无平方因子希尔伯特数

希尔伯特数指的是能表达为$4k+1$的正整数，其中整数$k\ge 0$。 我们定义无平方因子希尔伯特数为不能被除1之外的其它希尔伯特数的平方整除的希尔伯特数。例如，$117$是一个无平方因子希尔伯特数，它的质因数分解是$9\times 13$。 而$6237$则不是无平方因子希尔伯特数，因为它能够被$9^2$整除。$3969$也不是无平方因子希尔伯特数，因为它同时被$9^2$和${21}^2$整除。

一共有$2327192$个小于${10}^7$的无平方因子希尔伯特数。
有多少小于${10}^{16}$的无平方因子希尔伯特数？

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