Odd Untouchable Numbers
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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