Very Odd Numbers
A very odd number is a number which contains only odd digits and is divisible by $105$. Further each odd digit occurs an odd number of times.
Define $\Theta (n)$ be the $n$th very odd number, then $\Theta (1) = 1117935$ and $\Theta(10^3) = 11137955115$.
Find $\Theta(10^{16})$.