# untouchable number

An untouchable number is an integer $n$ for which there exists no integer $m$ such that

 $\left(\sum_{d|m}d\right)-m=n,$

, thus $n$ can’t be ”touched” by the sum of proper divisors of any other integer. Paul Erdős proved that there are infinitely many untouchable numbers.

Obviously no perfect number can be an untouchable number. Neither can any integer of the form $p+1$, where $p$ is a prime number. What is not so obvious is whether 5 is the only odd untouchable number, and the related question of whether 2 and 5 are the only prime untouchable numbers.

## References

• 1 P. Erdős, Über die Zahlen der Form $\sigma(n)-n$ und $n-\phi(n)$. Elem. Math. 28 (1973), 83–86.
Title untouchable number UntouchableNumber 2013-03-22 15:51:52 2013-03-22 15:51:52 PrimeFan (13766) PrimeFan (13766) 8 PrimeFan (13766) Definition msc 11A05