Maybe these random-like numbers came from hashing operations? For example: n-th key = truncate(SHA256(f((n-1)-th key))). It will be still hopeless if the process involves a strong passphrase though.
I think this would be a good way to do it. Then the author just needs to remember the seed and algorithm instead of remembering 256 private keys.
Although keeping track of 256 private keys is not that hard in the first place.
The algorithm would need to "skip over" any private keys in the generated sequence that had the undesirable result of a 0 in the most significant bit after the masking operation.