[seba_dos1]

The information is already included in the definition of untouchable number:

> a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer

There's only one sum of all the proper divisors for any given integer.

[casion]

> There's only one sum of all the proper divisors for any given integer.

This really seems to assume that uniqueness is an implicit property of each integer of such sums. I don't understand how you would know know that or how to discover that other than "you couldn't get the answers we're showing you unless you assumed that".

[PeterisP]

No, it only assumes that for every integer there exists a single, well defined set of "the proper divisors". Afterwards, summing all of them up is a trivial operation that can only possibly yield a single value.

[seba_dos1]

It says "the sum of all the proper divisors", not "the sum of a sequence consisting of all the proper divisors". It makes no sense to consider repetitions there (it gets easily reduced to absurd when you do), and it's already clear from the definition without even having to look at examples.