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untouchable numbers
A number    is called untouchable if it is not the sum of the proper divisors of any number, i.e. if it does not exist a number    such that  .

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.

ErdÅ‘s 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  .

The first untouchable numbers are 2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304 more terms

You can download a text file (untouchable_up1e6.txt) of 1.9 MB, containing a list of the 150232 untouchable numbers up to  .

Untouchable numbers can also be... (you may click on names or numbers and on + to get more values)