> Other hash applications (like hash tables) don't need that property

I'm unclear on what you guys mean by correlation, but if it means what I think it does I disagree with this. A hash table ideally has an uniform distribution regardless of the input, so any structured correlation with the input will harm this goal in real-world applications.

Python's hashes for small integers never collide.

Python 2.6.5 (r265:79063, Apr 16 2010, 13:57:41)

>>> map(hash, range(10))

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]