Distinct Lines

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 $D(10^{10})$. Give as your answer the first nine digits followed by the last nine digits.

Consider the lattice points lying on the plane \(x + y + z = N\). It is obvious that for every lattice point \(P(a, b, c)\) and the origin \(O(0, 0, 0)\), the line \(OP\) exists.

Every line \(OP\) can be expressed by \(O(a_1, a_2, a_3)\), where \(O\) is the origin, \((a_1, a_2, a_3)\) is a point that divides \(N\) exactly, and \(gcd(a_1, a_2, a_3) = 1\), with \(gcd\) being the greatest common divisor.

It is clear that the number of ways of choosing \((a_1, a_2, a_3)\) is \(D(N)\), so we have:

\[D(N) = \sum_{n=0}^{N} d(n)\], where \(d(n)\) denotes the number of positive divisors of \(n\).

We can apply the divisor function to calculate the number of factors of \(N\). Using the formula

\[d(n) = \prod_{i=1}^{r} (e_i + 1)\],

where \(e_i\) is the exponent in the prime factorization of \(n\).

For example, \(D(10^6)\) would be calculated by:

\[D(10^6) = D(2^6 \times 5^6)\]

Given the factorial nature of this problem, the numbers quickly become very large, and as such, this calculation must be done programmatically. You provided the value for \(D(1\,000\,000)\), which is \(831909254469114121\).

For the large number \(N = 10^{10}\), the calculation should be done programmatically. Given the nature of this problem, the calculation involves very large numbers and working with numbers this large analytically is not feasible in practice.

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