The numerical value is calculated as follows:
- A node with a 1 has a value which is 1 less than it otherwise would be.
- A (0) node with no children has the value 2.
- A (0) node with children of value n and m has the value of the sum of the nth and the mth prime number (counting 2 as prime number number one).
- Some numbers can be written more than one way. Find, for example, all ways of writing 10, 11 and 12.
- Suppose that f(x) is the number of ways in which the number x can be written. Find the asymptotic upper bound (the "ordo") for f(x).
- (hard) Is it really possible to write all positive integers this way? Prove it.