One of the problems of looking for patterns is matching known categories to the input. In your example, if prime numbers were not understood by the searcher, the pattern might be filed as "unknown".

On the other hand, if the searcher had a broad set of analysis tools [or access to Sloane's database], it might be able to build the category of prime numbers based on the given input (no divisors between 1 and N).

But the searcher must be careful to employ some simplification schemes (perhaps an Occam's Razor module?), because there are numerous formulas that will have those numbers on the curve, or as axis crossing points, etc.

I suspect that a good searcher will have to have or evolve infrastructure for category description and matching, as well as an engine for developing new categories and testing hypotheses.

Finally, I wasn't clear whether you meant "find the pattern in..." (such as the primes example), or "generalizations in time and/or space" (which is nearer the chess analogy). One involves induction, the other deduction [if I've got my head on straight today].

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