Odd Untouchable Numbers

Unproven Hypothesis

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

Progress

This would follow 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 aliquot parts 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 aliquot sums of $2$, $4$ and $8$ respectively.

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

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$

• Adams-Watters, Frank and Weisstein, Eric W.. "Untouchable Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/UntouchableNumber.html