| 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. |