TITLE: Numbers that are the sum of the squares of their prime factors
QUESTION [26 upvotes]: 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

REPLY [5 votes]: 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...