number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable
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)
Posted by Jer
on 2015-01-23 08:11:40