A smooth partition of the integer n is a set of positive integers a1,a2,…ak such that

Question

1. k is a positive integer,
2. a1≤a2≤⋯≤ak,
3. ∑ki=1ai=n, and
4. ak−a1≤1.
Determine how many smooth partitions there are of the integer 250.

Answer 1

250