Eccoci.

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 $a \gt 0$ 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 $f(10^{14})$.