Reachable Numbers
A positive integer will be called reachable if it can result from an arithmetic expression obeying the following rules:
- Uses the digits $1$ through $9$, in that order and exactly once each.
- Any successive digits can be concatenated (for example, using the digits $2$, $3$ and $4$ we obtain the number $234$).
- Only the four usual binary arithmetic operations (addition, subtraction, multiplication and division) are allowed.
- Each operation can be used any number of times, or not at all.
- Unary minusA minus sign applied to a single operand (as opposed to a subtraction operator between two operands) is not allowed.
- Any number of (possibly nested) parentheses may be used to define the order of operations.
For example, $42$ is reachable, since $(1 / 23) \times ((4 \times 5) - 6) \times (78 - 9) = 42$.
What is the sum of all positive reachable integers?