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).

Without computer assistance, finding all possible values for K below 1000 is not worth doing. As I don't plan of using a computer to iterate through all possible solutions, I only list a solution for each K from 5 to 14 [k={2,3,4} has already been given in the problem]. To demonstrate that a given K might have more than one solution, I provide one for K=14. K 5 (1,1,2,2,2) :=8 6 (1,1,1,1,2,6) :=12 7 (1,1,1,1,1,3,4) :=12 8 (1,1,1,1,1,2,2,3) :=12 9 (1,1,1,1,1,1,1,2,9) :=18 10 (1,1,1,1,1,1,1,1,4,4) :=16 11 (1,1,1,1,1,1,1,1,2,2,4) :=16 12 (1,1,1,1,1,1,1,1,1,1,2,12) :=24 13 (1,1,1,1,1,1,1,1,1,1,1,3,7) :=21 14 (1,1,1,1,1,1,1,1,1,1,1,2,2,5) :=20 14 (1,1,1,1,1,1,1,1,1,1,1,1,2,14) :=28