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Problem 821

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$123$-Separable

A set, $S$, of integers is called $123$-separable if $S$, $2S$ and $3S$ are disjoint. Here $2S$ and $3S$ are obtained by multiplying all the elements in $S$ by $2$ and $3$ respectively.

Define $F(n)$ to be the maximum number of elements of
$$(S\cup 2S\cup 3S)\cap \{1,2,3,\dots ,n\}$$
where $S$ ranges over all $123$-separable sets.

For example, $F(6)=5$ can be achieved with either $S=\{1,4,5\}$ or $S=\{1,5,6\}$.

You are also given $F(20)=19$.

Find $F({10}^{16})$.

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$123$可分离集合

若整数集合$S$满足$S$、$2S$和$3S$互不相交，则称其为$123$可分离集合。其中，$2S$和$3S$分别指将$S$中的所有元素乘以$2$或$3$所构成的集合。

考虑所有$123$可分离集合$S$，定义$F(n)$为集合
$$(S\cup 2S\cup 3S)\cap \{1,2,3,\dots ,n\}$$
中元素的最大数目。

例如，$F(6)=5$，对应的集合为$S=\{1,4,5\}$或$S=\{1,5,6\}$。

已知$F(20)=19$。

求$F({10}^{16})$。

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