For example, 2 is untouchable because for every prime , and it is easy to see that for every composite. On the contrary, 10 is not untouchable because the proper divisors of 14 are 1, 2, and 7, and 1 + 2 + 7 = 10.
Erdős has proved that there are infinitely many untouchable numbers.
If, as it is conjectured, every even number is the sum of two distinct primes, , , then 5 is the only odd untouchable number, since every larger odd number can be espressed as and thus be equal to the sum of the proper divisors of .
You can download a text file (untouchable_up1e6.txt) of 1.9 MB, containing a list of the 150232 untouchable numbers up to .