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Problem 586

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Binary Quadratic Form

The number $209$ can be expressed as $a^2+3ab+b^2$ in two distinct ways:

$$\begin{array}{rl}209& =8^2+3\cdot 8\cdot 5+5^2\\ 209& ={13}^2+3\cdot 13\cdot 1+1^2\end{array}$$

Let $f(n,r)$ be the number of integers $k$ not exceeding $n$ that can be expressed as $k=a^2+3ab+b^2$, with $a>b>0$ integers, in exactly $r$ different ways.

You are given that $f({10}^5,4)=237$ and $f({10}^8,6)=59517$.

Find $f({10}^{15},40)$.

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二元二次型

数$209$可以用两种方式表达为$a^2+3ab+b^2$：

$$\begin{array}{rl}209& =8^2+3\cdot 8\cdot 5+5^2\\ 209& ={13}^2+3\cdot 13\cdot 1+1^2\end{array}$$

记$f(n,r)$为不超过$n$且恰好可以用$r$种方式表达为$k=a^2+3ab+b^2$的整数$k$的数目，其中整数$a>b>0$。

已知$f({10}^5,4)=237$以及$f({10}^8,6)=59517$。

求$f({10}^{15},40)$。

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