Divisor Pairs

Let $S(n)$ be the number of pairs $(a,b)$ of distinct divisors of $n$ such that $a$ divides $b$.
For $n=6$ we get the following pairs: $(1,2),(1,3),(1,6),(2,6)$ and $(3,6)$. So $S(6)=5$.
Let $p_m#$ be the product of the first m prime numbers, so $p_2#=2∗3=6$.
Let $E(m,n)$ be the highest integer $k$ such that $2^k$ divides $S((p_m#)^n)$.
$E(2,1)=0$ since $2^0$ is the highest power of $2$ that divides $S(6)=5$.
Let $Q(n)=\sum_{i=1}^n E(904961,i)$
$Q(8)=2714886$.

Evaluate $Q(10^{12})$.