Unreachable Numbers

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Problem 718

Consider the equation $17^pa+19^pb+23^pc = n$ where $a$, $b$, $c$ and $p$ are positive integers, i.e. $a,b,c,p>0$.

For a given $p$ there are some values of $n > 0$ for which the equation cannot be solved. We call these unreachable values.

Define $G(p)$ to be the sum of all unreachable values of $n$ for the given value of $p$. For example $G(1) = 8253$ and $G(2)= 60258000$.

Find $G(6)$. Give your answer modulo $1\,000\,000\,007$.