I vaguely remember there is a notion of numbers rich in divisors, i.e. (number of divisors of N)/N is comparatively large. What's their name? Given a number M, how could I find such a number in its neighbourhood?
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This might be interesting: http://en.wikipedia.org/wiki/Highly_composite_number
Moreover if you care about sums: http://en.wikipedia.org/wiki/Abundant_number
You might be thinking about smooth numbers.
How to find such things? I suppose you could just multiply small numbers together until you are within the "neighbourhood". If you overshoot, take out some factors and replace them with other factors.