Higher order recursive sequence

Define a sequence $a_n$ as below: $a_1=17,a_{n+1}=\frac{a_n^4+6a_n^2+1}{4a_n(a_n^2+1)}(n\ge 1)$. Then $a_2=10657/2465$, $a_3=2134495165562497/1571545212141185$,...Obviously all numbers in this sequence are rational.

For a prime $p$ and a fully reduced fraction $a/b$, define $Q(a/b,p)$ to be the smallest positive $q$ for which $ab\equiv q(\mathrm{mod}p)$. For example $Q(3/5,109)=66$, because $5\cdot 66=330\equiv 3(\mathrm{mod}109)$ and $66$ is the smallest positive such number.

You are given $Q(a_2,1000000007)=877890477$, $Q(a_{10},1000000007)=16332768$.

Find the sum of $Q(a_{1234567891011121314},p)$ over all primes $p$ between 2000000000 and 2000100000.

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Thanks to czp for the idea.