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Problem 590

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Sets with a given Least Common Multiple

Let H($n$) denote the number of sets of positive integers such that the least common multiple of the integers in the set equals $n$.
E.g.:
The integers in the following ten sets all have a least common multiple of 6:
{2,3}, {1,2,3}, {6}, {1,6}, {2,6}, {1,2,6}, {3,6}, {1,3,6}, {2,3,6} and {1,2,3,6}.
Thus H(6)=10.

Let L($n$) denote the least common multiple of the numbers 1 through $n$.
E.g. L(6) is the least common multiple of the numbers 1,2,3,4,5,6 and L(6) equals 60.

Let HL($n$) denote H(L($n$)).
You are given HL(4)=H(12)=44.

Find HL(50000). Give your answer modulo 10⁹.

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给定最小公倍数的集合

记H($n$)表示各元素的最小公倍数为$n$的正整数集的数目。
例如：
下面这10个集合，其元素的最小公倍数都是6：
{2,3}，{1,2,3}，{6}，{1,6}，{2,6}，{1,2,6}，{3,6}，{1,3,6}，{2,3,6}和{1,2,3,6}。
因此H(6)=10。

记L($n$)为1到$n$的所有整数的最小公倍数。
例如，L(6)表示1，2，3，4，5，6的最小公倍数，因此L(6)等于60。

记HL($n$)为H(L($n$))。
已知HL(4)=H(12)=44。

求HL(50000)。将你的答案对10⁹取余。

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