An untouchable number is a positive integer that is not the sum of the proper divisors of any number.
The first few are 2, 5, 52, 88, 96, 120, 124, 146, ... (OEIS A005114).
Erdős has proven that there are infinitely many.
It is thought that 5 is the only odd untouchable number. This would follow from a very slightly stronger version of the Goldbach
conjecture, namely the conjecture that every even integer is the sum of two distinct primes. Suppose is an odd number greater than 7. Then by the conjecture, and so the proper divisors of , which are 1, , and , sum to , and so is not untouchable. 1, 3 and 7 are not untouchable, being
the sum of the proper divisors of 2, 4, and 8, respectively. That leaves 5 as the
only odd untouchable number (F. Adams-Watters, pers. comm., Aug. 4, 2006).
is the sum of two distinct primes. Suppose 
is an odd number greater than 7. Then 
by the conjecture, and so the proper divisors of 
, which are 1, 
, and 
, sum to 
, and so 
is not untouchable. 1, 3 and 7 are not untouchable, being
the sum of the proper divisors of 2, 4, and 8, respectively. That leaves 5 as the
only odd untouchable number (F. Adams-Watters, pers. comm., Aug. 4, 2006).
