Distinct Lines

Problem Statement

Consider all lattice points $(a,b,c)$ with $0 \le a,b,c \le N$.

From the origin $O(0,0,0)$ all lines are drawn to the other lattice points.
Let $D(N)$ be the number of distinct such lines.

You are given that $D(1\,000\,000) = 831909254469114121$.

Find $D(2^N)$.

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