An untouchable number is a positive integer that is not the sum of the positive proper
divisors of any positive integer. The first
few are 2, 5, 52, 88, 96, 120, 124, 146, ... (OEIS A005114).
Erdős (1973, p. 83) proved 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
number
is the sum of two distinct primes.
Suppose
is an odd number greater than 7. Then by the conjecture, and so the positive 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 positive proper divisors of 2, 4, and 8,
respectively. Under this conjecture, 5 is therefore the only odd
untouchable number (Erdős 1973, p. 84; 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 positive 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 positive proper divisors of 2, 4, and 8,
respectively. Under this conjecture, 5 is therefore the only odd
untouchable number (Erdős 1973, p. 84; F. Adams-Watters, pers. comm.,
Aug. 4, 2006).
und 
."
Elem. Math. 28, 83-86, 1973.