Mapping Unique 16-Digit numeric ID to Unique Alphanumeric ID

In a project I'm working on I need to generate 16 character long unique IDs, consisting of 10 numbers plus 26 uppercase letters (only uppercase). They must be guaranteed to be universally unique, with zero chance of a repeat ever. The IDs are not stored forever, an ID is thrown out of the database after a period of time and a new unique ID must be generated. The IDs can never repeat with the thrown out ones either. So randomly generating 16 digits and checking against a list of previously generated IDs is not an option because there is no comprehensive list of previous IDs. Also, UUID will not work because the IDs must be 16 digits in length.

Right now I'm using 16-Digit Unique IDs, that are guaranteed to be universally unique every time they're generated (I'm using timestamps to generate them + unique server ID). However, I need the IDs to be difficult to predict, and using timestamps makes them easy to predict. So what I need to do is map the 16 digit numeric IDs that I have into the larger range of 10 digits + 26 letters without losing uniqueness. I need some sort of hashing function that maps from a smaller range to a larger range, guaranteeing a one-to-one mapping so that the unique IDs are guaranteed to stay unique after being mapped. I have searched and so far have not found any hashing or mapping functions that are guaranteed to be collision-free, but one must exist if I'm mapping to a larger space. Any suggestions are appreciated.