Posted by **best** on Tuesday, June 18, 2013 at 11:32pm.

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.

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