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
Canonical name	UntouchableNumber
Date of creation	2013-03-22 15:51:52
Last modified on	2013-03-22 15:51:52
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	8
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A05