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A smooth partition of the integer n is a set of positive integers a 1 ,a 2 ,…a k such that
1. k is a positive integer,
2. a 1 ≤a 2 ≤⋯≤a k ,
3. ∑ k i=1 a i =n, and
4. a k −a 1 ≤1.
Determine how many smooth partitions there are of the integer 250.
Note: The definition given above is not standard terminology.