An untouchable number is a positive integer that is not the sum of the proper divisors of any number(see OEIS A005114).
It is thought that 5 is the only odd untouchable number.
Why?
The Goldbach conjecture states every even number starting with 4 can be written as the sum of two primes. This is widely believed to be true. A slightly stronger version states that every even number starting with 8 can be written as the sum of two
distinct odd primes.
Assume the stronger version is in fact true. Then every odd number k starting with 9 can be written as k=1+p+q where p and q are distinct odd primes. Then let x be the number x=p*q. Then the sum of the proper divisors of x is k.
This leaves 1, 3, and 7 to check by hand. 1 is touchable by any prime. 3 is touchable by 4 (1+2). 7 is touchable by 8 (1+2+4). This leaves 5 as the only untouchable odd number.
Goldbach Solution