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Problem of the Week 1239
Uniquely Unique Partitions
For a partition p of a positive integer N into k positive integer parts, the "graph" of p has a vertex for each entry in p and an edge between two vertices if the corresponding entries of p have a common divisor greater than 1.
Example: The graph of the partition 10 = 4 + 2 + 2 + 1 + 1 has 5 vertices and 3 edges. A partition of N is "recoverable" if it is determined by N and the number of edges in its graph. For example,
What is the largest N for which there is a unique recoverable partition of N?
Source: F. Barrera (Bogota, Colombia), B. Recaman (Bogota, Colombia), and S. Wagon.