## Divisor Pairs

### Problem 561

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 $Q(10^{12})$.