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Problem 735

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Divisors of $2n^2$

Let $f(n)$ be the number of divisors of $2n^2$ that are no greater than $n$. For example, $f(15)=8$ because there are $8$ such divisors: $1,2,3,5,6,9,10,15$. Note that $18$ is also a divisor of $2\times 15^2$ but it is not counted because it is greater than $15$.

Let $\displaystyle F(N) = \sum_{n=1}^N f(n)$. You are given $F(15)=63$, and $F(1000)=15066$.

Find $F(10^{12})$.

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$2n^2$的约数

记$f(n)$为$2n^2$的不超过$n$的约数数目。例如，$f(15)=8$因为有$8$个满足条件的约数：$1,2,3,5,6,9,10,15$。注意尽管$18$也是$2\times 15^2$的约数，但是因为它大于$15$因此不被计算在内。

记$\displaystyle F(N) = \sum_{n=1}^N f(n)$。已知$F(15)=63$，$F(1000)=15066$。

求$F(10^{12})$。

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