Problem 974

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 $\Theta(10^{16})$.

若一个数仅包含奇数数字、能被$105$整除且每个奇数数字出现的次数均为奇数，则称其为非常奇数。

定义$\Theta (n)$为第$n$个非常奇数，则$\Theta (1) = 1117935$，$\Theta(10^3) = 11137955115$。

求$\Theta(10^{16})$。