Euler's Product formula upside down

This model shows how if you put $1, 2, 2^2, 2^3, ...$, along one axis, $1,3, 3^2, 3^3, ...$ along another and $1, 5, 5^2, 5^3, ...$ along a third and then fill out the 3D table by multiplying everything, you end up getting all possible numbers made from 2s, 3s and 5s.

If you imagine this with a dimension for every prime number you get the formula $\prod{\sum{p^k}} = \sum{n}$