Inmathematics, anuntouchable number is a positiveinteger that cannot be expressed as thesum of all theproper divisors of any positive integer. That is, these numbers are not in the image of thealiquot sum function. Their study goes back at least toAbu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 and 5 are untouchable.[1]
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 + 3 = 4.
The number 5 is untouchable, as it is not the sum of the proper divisors of any positive integer: 5 = 1 + 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 + 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 + 2 + 3 = 6.
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 theGoldbach conjecture, since the sum of the proper divisors ofpq (withp,q distinct primes) is 1 +p +q. Thus, if a numbern can be written as a sum of two distinct primes, thenn + 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,,, so only 5 can be an odd untouchable number.[2] Thus it appears that besides 2 and 5, all untouchable numbers arecomposite numbers (since except 2, all even numbers are composite). Noperfect number is untouchable, since, at the very least, it can be expressed as the sum of its own properdivisors. Similarly, none of theamicable numbers orsociable numbers are untouchable. Also, none of theMersenne numbers are untouchable, sinceMn = 2n − 1 is equal to the sum of the proper divisors of 2n.
No untouchable number is one more than aprime number, since ifp is prime, then the sum of the proper divisors ofp2 is p + 1. Also, no untouchable number is three more than a prime number, except 5, since ifp is an odd prime then the sum of the proper divisors of 2p is p + 3.
There are infinitely many untouchable numbers, a fact that was proven byPaul Erdős.[3] According to Chen & Zhao, theirnatural density is at least d > 0.06.[4]
^The stronger version is obtained by adding to the Goldbach conjecture the further requirement that the two primes be distinct—seeAdams-Watters, Frank & Weisstein, Eric W."Untouchable Number".MathWorld.
^P. Erdos, Über die Zahlen der Form und. Elemente der Math. 28 (1973), 83-86
^Yong-Gao Chen and Qing-Qing Zhao, Nonaliquot numbers, Publ. Math. Debrecen 78:2 (2011), pp. 439-442.
