---

Problem 821

---

$123$-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
$$(S\cup 2S \cup 3S)\cap \{1,2,3,\ldots,n\}$$
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 $F(10^{16})$.

---

$123$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
$$(S\cup 2S \cup 3S)\cap \{1,2,3,\ldots,n\}$$
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$F(10^{16})$。

---