Numbers that are the sum of the squares of their prime factors

Question

A number which is equal to the sum of the squares of its prime factors with multiplicity:

• $16=2^2+2^2+2^2+2^2$
• $27=3^2+3^2+3^2$

Are these the only two such numbers to exist?

There has to be an easy proof for this, but it seems to elude me.

Thanks

Answer 1

Here is a suggestion:

For a start one could investigate under which assumptions about the sizes of $n$ and real variables $x_k\geq2$ $\>(1\leq k\leq n)$ an equality $$\prod_{k=1}^n x_k=\sum_{k=1}^n x_k^2$$ is at all possible.

Answer 2

Giorgos Kalogeropoulos has found 3 such numbers, each having more than 100 digits.
You can find these numbers if you follow the links in the comments of OEIS A339062 & A338093

or here https://www.primepuzzles.net/puzzles/puzz_1019.htm

So, such numbers exist! It is an open question if there are infinitely many of them...