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Problem 858

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LCM

Define $G(N) = \sum_S \operatorname{lcm}(S)$ where $S$ ranges through all subsets of ${1, \dots, N}$ and $\operatorname{lcm}$ denotes the lowest common multiple. Note that the $\operatorname{lcm}$ of the empty set is $1$.

You are given $G(5) = 528$ and $G(20) = 8463108648960$.

Find $G(800)$. Give your answer modulo $10^9 + 7$.

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最小公倍数

定义$G(N) = \sum_S \operatorname{lcm}(S)$，其中$S$取遍${1, \dots, N}$的所有子集，$\operatorname{lcm}$表示最小公倍数。注意对空集取$\operatorname{lcm}$的结果为$1$。

已知$G(5) = 528$，$G(20) = 8463108648960$。

求$G(800)$，并对$10^9 + 7$取余作为你的答案。

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