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 $\phi(a)-\phi(b)$, where $\phi$ 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 $715827883$.
Solution
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