## Sets with a given Least Common Multiple

### Problem 590

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 10^{9}.