Smallest Number which is Sum of 4 Triples with Equal Products

Theorem
The smallest positive integer which is the sum of $4$ distinct ordered triples, each of which has the same product, is $118$:

Computational proof
The following python script finds the maximum number of ways to represent each positive integer as sums of triples with the same product:

This outputs:

This confirms the statement that $118$ is the smallest positive integer that satisfies the criterion.

Also see

 * Smallest Integer which is Product of 4 Triples all with Same Sum