Sets with a given Least Common Multiple (noch nicht übersetzt)

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.