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# 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.

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new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier