A Good and SIMPLE Measure of Randomness

Your question answers itself. "If I were to pass the first 100,000 digits of Pi to the function, it should give a number very close to 1", except the digits of Pi are not random numbers so if your algorithm does not recognise a very specific sequence as being non-random then it's not very good.

The problem here is there are many types of non random-ness:- eg. "121,351,991,7898651,12398469018461" or "33,27,99,3000,63,231" or even "14297141600464,14344872783104,819534228736,3490442496" are definitely not random.

I think what you need to do is identify the aspects of randomness that are important to you- distribution, distribution of digits, lack of common factors, the expected number of primes, Fibonacci and other "special" numbers etc. etc.

PS. The Quick and Dirty (and very effective) test of randomness does the file end up roughly the same size after you gzip it.

You could try to zip-compress the sequence. The better you succeed the less random the sequence is.

Thus, heuristic randomness = length of zip-code/length of original sequence

It can be done this way:

CAcert Research Lab does a Random Number Generator Analysis.

Their results page evaluates each random sequence using 7 tests (Entropy, Birthday Spacing, Matrix Ranks, 6x8 Matrix Ranks, Minimum Distance, Random Spheres, and the Squeeze). Each test result is then color coded as one of "No Problems", "Potentially deterministic" and "Not Random".

So a function can be written that accepts a random sequence and does the 7 tests. If any of the 7 tests are "Not Random" then the function returns a 0. If all of the 7 tests are "No Problems", then it returns a 1. Otherwise, it can return some number in-between based on how many tests come in as "Potentially Deterministic".

The only thing missing from this solution is the code for the 7 tests.