Let's start with an easy problem : What three positive integers have a sum equal to their product?
answer: (1,2,3), of course.

This puzzle can easily be transformed into a D4 problem:

For what values of k will the question "What k positive integers have a sum equal to their product?" have only one unique set of integers for an answer?
Clearly for k=2 the answer is unique: (2,2) and so it is for k=4: (1,1,2,4).

List all other values of k below 1000.