Inspired by Nihar Mahajan's featured problem

In each of the cases below, there exists only one solution whereby the sum of nn positive integers equals the product.

  • For n=2n=2, we have 2+2=2×22+2=2\times 2.
  • For n=3n=3, we have 1+2+3=1×2×31+2+3=1\times2\times3.
  • For n=4n=4, we have 1+1+2+4=1×1×2×41+1+2+4=1\times 1\times2\times4.

If solution(s) exist for n=5,n=5, find the sum of all possible sums (or equivalently, products).

If such a solution does not exist, submit your answer as 0.


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