Question 88
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 $1 \leq a\lt 10$ is a decimal number.
If your answer is written in this form, your score will be $\max \left\{0,25-\left\lfloor 100\left|\log _{10}(A / N)\right|\right\rfloor\right\}$, where $N=a \cdot 10^{b}$ is your answer to this problem and $A$ is the actual answer. Otherwise, your score will be zero.
$$
\sum_{n=1}^{10^{6}} n f(n)
$$
Write your answer in the form $a \cdot 10^{b}$, where $b$ is an integer and $1 \leq a\lt 10$ is a decimal number.
If your answer is written in this form, your score will be $\max \left\{0,25-\left\lfloor 100\left|\log _{10}(A / N)\right|\right\rfloor\right\}$, where $N=a \cdot 10^{b}$ is your answer to this problem and $A$ is the actual answer. Otherwise, your score will be zero.