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# Problem 561

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 $Q(10^{12})$.