HMMT 2015 U34

For an integer $n$, let $f(n)$ denote the number of pairs $(x, y)$ of integers such that $x^{2}+x y+y^{2}=n$. Compute the sum

$$\sum_{n=1}^{10^{6}} n f(n)$$

Write your answer in the form $a \cdot 10^{b}$, where $b$ is an integer and $1 \leq a<10$ is a decimal number. If your answer is written in this form, your score will be $\left.\max \left\{0,25-\left\lfloor 100\left|\log _{10}(A / N)\right|\right\rfloor\right)\right\}$, where $N=a \cdot 10^{b}$ is your answer to this problem and $A$ is the actual answer. Otherwise, your score will be zero.