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).

Prove:
i. There is no untouchable number that is one more than a prime.
ii. Except number 5, there is no untouchable number that is
three more than a prime.

Let p be a prime number.

Is there a numbers whose proper divisors sum to p+1?  Yes.
The divisors of p² are 1 and p.

Is there a numbers whose proper divisors sum to p+3?  Yes.
Unless p=2, the divisors of 2p are 1,2, and p.

The second can be expanded:
There is no untouchable number that is four more than a prime.  Consider the divisors of 3p.
(7 is touched by 8)
etc.

Posted by Jer on 2015-01-23 08:11:40