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Problem 590

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.

给定最小公倍数的集合

{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}。