Untouchable number

Their study goes back at least to Abu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 and 5 are untouchable.

It would follow from a slightly stronger version of the Goldbach conjecture, since the sum of the proper divisors of pq (with p, q distinct primes) is 1 + p + q.

No perfect number is untouchable, since, at the very least, it can be expressed as the sum of its own proper divisors.

Also, none of the Mersenne numbers are untouchable, since Mn = 2n − 1 is equal to the sum of the proper divisors of 2n-1.

There are infinitely many untouchable numbers, a fact that was proven by Paul Erdős.

If we draw an arrow pointing from each positive integer to the sum of all its proper divisors, there will be no arrow pointing to untouchable numbers like 2 and 5.