For a positive integer $n$ construct a graph using all the divisors of $n$ as the vertices. An edge is drawn between $a$ and $b$ if $a$ is divisible by $b$ and $a/b$ is prime, and is given weight $\varphi (a)-\varphi (b)$, where $\varphi$ is the Euler totient function.
Define $t(n)$ to be the total weight of this graph.
The example below shows that $t(45)=52$

Let $T(N)={\displaystyle \sum _{n=1}^{N}t(n)}$. You are given $T(10)=26$ and $T({10}^2)=5282$.

Find $T({10}^{12})$. Give your answer modulo $715827883$.