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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.

> 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".

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.

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.

It's somewhat implied by the first example on the Untouchable number page.

> The number 4 is not untouchable as it is equal to the sum of the proper divisors of 9: 1 + 3 = 4.

If repeated divisors were counted, that sum would be 7, not 4.

Another way to realize it would be to notice that 1 is a very special case if repetition counted. Would 1 be represented as a proper divisor infinite times? that would make every sum infinite. Or would there be a special case for why 1 is only counted once, but other numbers can be counted multiple times? or do you just exclude 1 from being a proper divisor?

If all the sums were infinite it wouldn't be interesting.. I suspect with the other 2 types of rules, almost none of the numbers would be "untouchable", which also makes it not interesting.

Count 1 only once and allowing repetition is this way. I found that 2 is "untouchable" since sum(1) = 1, sum(2) = 3, sum (3) = 4, sum(4) = 5. It's easy to see that every other sum will be larger than two.

All other even number is "touchable": for even n, take the lowest prime less than n and then multiply it by 2 a few times. With the 2s and the odd prime and the extra 1, it can be equal to n.

Similarly all other odd numbers are "touchable": for odd n, take the lowest prime p less than n and either multiply it by 2 if p = n - 2, or multiply it by 3 to make it even, then multiply by 2 one or more times and with the 1 it becomes odd n.

I suspect excluding 1 in the sum but allowing repetition is similarly trivial.

These comments ITT are great for understanding. I wish someone would update the Wikipedia article with a good explanation based on the comments ITT, so that future readers of the Wikipedia article could benefit from this information

That someone could be you!

One way you could have figured that out was realizing that every number has 1 as a proper divisor, so even number would be trivially touchable if repetition was allowed

Don’t feel bad. The wiki page is very poorly written like pretty much all the math pages.

I'm curious about this, another comment in the thread expressed the same opinion about math pages on wiki, while I've always heard the opposite opinion among mathematicians. Could you say a bit more on what makes pretty much all of the math pages on wiki very poorly written?

They're generally written in a way where it's not helpful to teach yourself about the topic, but is helpful to refresh yourself if you've previously learned about it elsewhere.

I think you missed the (implicit) definition of “the sum”: If someone asked you “what is the sum of all the proper divisors of 8?”, would you say it's 1+2+4=7, or would you say, “well, it could be either 1+2+4=7, or 1+2+2+4=9, or 1+1+1+2+2+4+4+4=19, or…”?

Anyway, yes, reading such definitions comes with experience (what's called “mathematical maturity”: https://en.wikipedia.org/wiki/Mathematical_maturity) — you have to assume that everything being stated is well-defined and makes sense, that every word matters (e.g. you can't ignore the word “all” in “the sum of all proper divisors”, nor can you ignore the word “the”!), and then, if that doesn't work, try to see if alternative definitions make sense. Also, sometimes things are stated in two different ways; that's usually helpful. In this case, given that the first two sentences are:

> An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself). That is, these numbers are not in the image of the aliquot sum function.

Note the “That is”: If the first way of stating the definition doesn't seem clear / doesn't seem to fit the example that immediately follows, then try the second sentence, specifically follow the “aliquot sum” link to https://en.wikipedia.org/w/index.php?title=Aliquot_sum&oldid... which makes it very clear:

> For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are 1, 2, 3, 4, and 6, so the aliquot sum of 12 is 16 i.e. (1 + 2 + 3 + 4 + 6).

[…]

> The untouchable numbers are the numbers that are not the aliquot sum of any other number.