Untouchable Numbers

I’ve dealt with untouchable numbers before but only in passing. I’ve never devoted an entire post to the topic. In my Mathematical Meandering blog on Blogger, I mentioned this category of numbers in two posts: one titled The Connectivity of Numbers and the other Mathematical Properties of 2022. My diurnal age today happens to be 26708 and this number turns out to be untouchable, meaning that there are no numbers whose sum of aliquot parts is equal to this number.

26798 is the 3470th untouchable number, meaning that the frequency of such numbers over this range is about 13%. As the range is extended, it has been shown that this density remains at least greater than 6% and Paul Erdos has shown that there are infinitely many untouchable numbers. These numbers are listed in OEIS A005114 and the sequence begins:

2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658, …

It should be noted that a number cannot be untouchable if it is one more than a prime number \( p \) because then it would be the sum of the aliquot parts of \( p^2 \). Similarly, if a number is three more a prime number \( p \), it cannot be untouchable because then it would be the aliquot sum of \( 2p \). More formally, we can say that untouchable numbers are those numbers that are not in the range of the aliquot sum function \( s(n) \) where \( n \) is any positive integer and \( d \) represents its divisors:$$s(n)=\sum_{d|n, d \neq n} \! \! \! d $$The conjecture is that 5 is the only odd untouchable number but this has not been proven. If true, then all untouchable numbers are composite except 2. Writing a program to generate untouchable numbers is not as simple as it might seem because some quite large numbers can have a relatively small aliquot sum. Below is an SageMath algorithm that will generate the untouchable numbers in the range from 26798 to 26750 (permalink). If the search range drops much below 500,000, “false positives” will begin to appear. Feel free to experiment.