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 20150123 08:11:40 