I had a think about this and came to some conclusions. The likelihood of finding a continuous series of numbers in an infinite series obviously diminishes with the length of the data you wish to "find" in the sequence. The other component of this scheme is the offset, you need to know at which digit you have to start to read the "file" or data sequence. Finding that start digit is the first key to recording your "key" or initial number in the data file. Then you have to know the last number as well. There are some algorithms for computing arbitrary digits, for instance in pi, but no universal scheme for finding that sequence exists for all infinite series, repeating or non.
So I propose an easier way. What if you broke your "file" or data you wish to encode into smaller pieces, maybe just bytes or 2-4 byte groups. That would make "finding" the proper numbers much less computationally intensive. On the downside, you'd have a larger "key set" or start and stop digits for each "file" you wish to save, but it would make storing the file much easier. Maybe to save space you can specify longer groups if you're willing to spend the time to find each start / stop digit for each group.
