# 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})\$.