A Gaussian integer is a number $z=a+bi$ where $a$, $b$ are integers and $i^2=-1$.
Gaussian integers are a subset of the complex numbers, and the integers are the subset of Gaussian integers for which $b=0$.

A Gaussian integer unit is one for which $a^2+b^2=1$, i.e. one of $1,i,-1,-i$.
Let's define a proper Gaussian integer as one for which $a>0$ and $b\ge 0$.

A Gaussian integer $z_1=a_1+b_{1}i$ is said to be divisible by $z_2=a_2+b_{2}i$ if $z_3=a_3+b_{3}i=z_1/z_2$ is a Gaussian integer.
$\frac{z_1}{z_2}=\frac{a_1+b_{1}i}{a_2+b_{2}i}=\frac{(a_1+b_{1}i)(a_2-b_{2}i)}{(a_2+b_{2}i)(a_2-b_{2}i)}=\frac{a_1{a}_2+b_1{b}_2}{a_2^2+b_2^2}+\frac{a_2{b}_1-a_1{b}_2}{a_2^2+b_2^2}i=a_3+b_{3}i$
So, $z_1$ is divisible by $z_2$ if $\frac{a_1{a}_2+b_1{b}_2}{a_2^2+b_2^2}$ and $\frac{a_2{b}_1-a_1{b}_2}{a_2^2+b_2^2}$ are integers.
For example, $2$ is divisible by $1+i$ because $2/(1+i)=1-i$ is a Gaussian integer.

A Gaussian prime is a Gaussian integer that is divisible only by a unit, itself or itself times a unit.
For example, $1+2i$ is a Gaussian prime, because it is only divisible by $1$, $i$, $-1$, $-i$, $1+2i$, $i(1+2i)=i-2$, $-(1+2i)=-1-2i$ and $-i(1+2i)=2-i$.
$2$ is not a Gaussian prime as it is divisible by $1+i$.

A Gaussian integer can be uniquely factored as the product of a unit and proper Gaussian primes.
For example $2=-i(1+i{)}^2$ and $1+3i=(1+i)(2+i)$.
A Gaussian integer is said to be squarefree if its prime factorization does not contain repeated proper Gaussian primes.
So $2$ is not squarefree over the Gaussian integers, but $1+3i$ is.
Units and Gaussian primes are squarefree by definition.

Let $f(n)$ be the count of proper squarefree Gaussian integers with $a^2+b^2\le n$.
For example $f(10)=7$ because $1$, $1+i$, $1+2i$, $1+3i=(1+i)(2+i)$, $2+i$, $3$ and $3+i=-i(1+i)(1+2i)$ are squarefree, while $2=-i(1+i{)}^2$ and $2+2i=-i(1+i{)}^3$ are not.
You are given $f({10}^2)=54$, $f({10}^4)=5218$ and $f({10}^8)=52126906$.

Find $f({10}^{14})$.