Odd Untouchable Numbers

Unproven Hypothesis
It is highly likely that $5$ is the only odd untouchable number.

Progress
This would from a slightly stronger form of Goldbach's Conjecture:
 * Every even integer $n > 6$ is the sum of two distinct primes.

Let $2 n + 1$ be an odd integer number greater than $7$.

Then by the conjecture:
 * $2 n = p + q$

The proper divisors of $p q$ are $1$, $p$ and $q$.

Then:
 * $1 + p + q = 2 n + 1$

and so $2 n + 1$ is not untouchable.

Then we have that $1$, $3$ and $7$ are not untouchable, as they are the sum of the proper divisors of $2$, $4$ and $8$ respectively.

That leaves $5$ as the only odd untouchable number.