Eccoci.

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 $10^9$.