View source for Untouchable number

{{Short description|Number that cannot be written as an aliquot sum}}
In [[mathematics]], an '''untouchable number''' is a positive [[integer]] that cannot be expressed as the [[summation|sum]] of all the [[proper divisor]]s of any positive integer. That is, these numbers are not in the image of the [[aliquot sum]] function. Their study goes back at least to [[Abu Mansur al-Baghdadi]] (circa 1000 AD), who observed that both 2 and 5 are untouchable.<ref>{{citation
 | last = Sesiano | first = J.
 | issue = 3
 | journal = Archive for History of Exact Sciences
 | jstor = 41133889
 | mr = 1107382
 | pages = 235–238
 | title = Two problems of number theory in Islamic times
 | volume = 41
 | year = 1991
 | doi = 10.1007/BF00348408| s2cid = 115235810
 }}</ref>

==Examples==
[[File:Aliquot sums and untouchable numbers.pdf|thumb|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.]]

* The number 4 is not untouchable, as it is equal to the sum of the proper divisors of 9: 1&nbsp;+&nbsp;3&nbsp;=&nbsp;4.
* The number 5 is untouchable, as it is not the sum of the proper divisors of any positive integer: 5&nbsp;=&nbsp;1&nbsp;+&nbsp;4 is the only way to write 5 as the sum of distinct positive integers including 1, but if 4 divides a number, 2 does also, so 1&nbsp;+&nbsp;4 cannot be the sum of all of any number's proper divisors (since the list of factors would have to contain both 4 and 2).
* The number 6 is not untouchable, as it is equal to the sum of the proper divisors of 6 itself: 1&nbsp;+&nbsp;2&nbsp;+&nbsp;3&nbsp;=&nbsp;6.

The first few untouchable numbers are
:[[2 (number)|2]], [[5 (number)|5]], [[52 (number)|52]], [[88 (number)|88]], [[96 (number)|96]], [[120 (number)|120]], [[124 (number)|124]], [[146 (number)|146]], [[162 (number)|162]], [[188 (number)|188]], [[206 (number)|206]], [[210 (number)|210]], [[216 (number)|216]], [[238 (number)|238]], [[246 (number)|246]], [[248 (number)|248]], [[262 (number)|262]], [[268 (number)|268]], [[276 (number)|276]], [[288 (number)|288]], [[290 (number)|290]], 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ... {{OEIS|id=A005114}}.

==Properties==
{{unsolved|mathematics|Are there any odd untouchable numbers other than 5?}}
The number 5 is believed to be the only odd untouchable number, but this has not been proven.  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''. Thus, if a number ''n'' can be written as a sum of two distinct primes, then ''n'' + 1 is not an untouchable number. It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 7 is an untouchable number, and <math>1=\sigma(2)-2</math>, <math>3=\sigma(4)-4</math>, <math>7=\sigma(8)-8</math>, so only 5 can be an odd untouchable number.<ref>The stronger version is obtained by adding to the Goldbach conjecture the further requirement that the two primes be distinct—see {{MathWorld|urlname=UntouchableNumber|title=Untouchable Number|author=Adams-Watters, Frank|author2=Weisstein, Eric W.|name-list-style=amp}}</ref> Thus it appears that besides 2 and 5, all untouchable numbers are [[composite number]]s (since except 2, all even numbers are composite). No [[perfect number]] is untouchable, since, at the very least, it can be expressed as the sum of its own proper [[divisor]]s. Similarly, none of the [[amicable number]]s or [[sociable number]]s are untouchable. Also, none of the [[Mersenne number]]s are untouchable, since ''M''<sub>''n''</sub> = 2<sup>''n''</sup> − 1 is equal to the sum of the proper divisors of 2<sup>''n''</sup>.

No untouchable number is one more than a [[prime number]], since if ''p'' is prime, then the sum of the proper divisors of ''p''<sup>2</sup> is&nbsp;''p''&nbsp;+&nbsp;1. Also, no untouchable number is three more than a prime number, except 5, since if ''p'' is an odd prime then the sum of the proper divisors of 2''p'' is&nbsp;''p''&nbsp;+&nbsp;3.

==Infinitude==
There are infinitely many untouchable numbers, a fact that was proven by [[Paul Erdős]].<ref>P. Erdos, Über die Zahlen der Form <math>\sigma(n)-n</math> und <math>n-\phi(n)</math>. Elemente der Math. 28 (1973), 83-86</ref> According to Chen & Zhao, their [[natural density]] is at least d > 0.06.<ref>Yong-Gao Chen and Qing-Qing Zhao, Nonaliquot numbers, Publ. Math. Debrecen 78:2 (2011), pp. 439-442.</ref>

==See also==
* [[Aliquot sequence]]
* [[Nontotient]]
* [[Noncototient]]
* [[Weird number]]

== References ==
{{Reflist}}
* [[Richard K. Guy]], ''Unsolved Problems in Number Theory'' (3rd ed), [[Springer Verlag]], 2004 {{ISBN|0-387-20860-7}}; section B10.

== External links ==
*{{OEIS el|sequencenumber=A070015|name=Least m such that sum of aliquot parts of m equals n or 0 if no such number exists|formalname=Least m such that the sum of the aliquot parts of m (A001065) equals n, or 0 if no such number exists}}

{{Classes of natural numbers}}
{{Divisor classes}}

[[Category:Arithmetic dynamics]]
[[Category:Divisor function]]
[[Category:Integer sequences]]