Consider the values of ${\log }_2(8)$, ${\log }_4(64)$ and ${\log }_3(27)$. All three are equal to $3$.

Generally, the function $f(m,n)={\log }_m(n)$ over integers $m,n\ge 2$ has the property that
$f(m_1,n_1)=f(m_2,n_2)$ if

1. $m_1=a^e,n_1=a^f,m_2=b^e,n_2=b^f$ for some integers $a,b,e,f$ or
2. $m_1=a^e,n_1=b^e,m_2=a^f,n_2=b^f$ for some integers $a,b,e,f$

We call a function $g(m,n)$ over integers $m,n\ge 2$ proto-logarithmic if

• $g(m_1,n_1)=g(m_2,n_2)$ if any integers $a,b,e,f$ fulfilling 1. or 2. can be found
• and $g(m_1,n_1)\ne g(m_2,n_2)$ if no integers $a,b,e,f$ fulfilling 1. or 2. can be found.

Let $D(N)$ be the number of distinct values that any proto-logarithmic function $g(m,n)$ attains over $2\le m,n\le N$.
For example, $D(5)=13$, $D(10)=69$, $D(100)=9607$ and $D(10000)=99959605$.

Find $D({10}^{18})$, and give the last $9$ digits as answer.

Note: According to the four exponentials conjecture the function ${\log }_m(n)$ is proto-logarithmic.
While this conjecture is yet unproven in general, ${\log }_m(n)$ can be used to calculate $D(N)$ for small values of $N$.