Mathematical Meanderings: More on Anti-divisors

Back in February of 2016, I made a post about anti-divisors but I've posted nothing about them since. However, I was reminded of them once again because today I turned 26264 days old and one of the properties of this number is that it's a member of OEIS:

A109351

Numbers whose anti-divisors sum to a perfect cube.

The anti-divisors of 26264 are 3, 16, 112, 784, 1072, 7504 and 17509. My original post explains how anti-divisors are determined so I won't repeat all of that here but essentially:

if the anti-divisor \(k\) of the number \(n\) is even then \( n \equiv \displaystyle \frac{k}{2} \mod{k} \)
if the anti-divisor \(k\) of the number \(n\) is odd then \( n \equiv \displaystyle \frac{k+1}{2} \mod{k} \text{ or } \displaystyle \frac{k-1}{2} \mod{k} \)

In the case of 26264 and its anti-divisors, we find that:

\( 26264 \equiv 2 \mod{3} \text{ and } 2=\displaystyle \frac{3+1}{2}\)
\( 26264 \equiv 8 \mod{16} \text{ and } 8=\displaystyle \frac{16}{2}\)
\( 26264 \equiv 56 \mod{112} \text{ and } 56=\displaystyle \frac{112}{2} \)
\( 26264 \equiv 397 \mod{784} \text{ and } 397=\displaystyle \frac{784}{2}\)
\( 26264 \equiv 536 \mod{1072} \text{ and } 536=\displaystyle \frac{1072}{2}\)
\( 26264 \equiv 3752 \mod{7504} \text{ and } 3752=\displaystyle \frac{7504}{2}\)
\( 26264 \equiv 8755 \mod{17509} \text{ and } 8755= \displaystyle \frac{17509+1}{2}\)

There are lots of interesting facts about anti-divisors listed on this source:

Every integer \(n\) has a largest anti-divisor, and this is at approximately at 2/3rds of \(n\).

Every number has a unique set of anti-divisors.

Anti-primes (integers with only one anti-divisor) are rare and include 3, 4, 6, 96 and 393216. These numbers form OEIS A066466.

Anti-perfect numbers are integers such that the sum of its anti-divisors equals the original integers. Figure 1 shows a list of the initial members.

Figure 1

The anti-perfect numbers form OEIS A073930:

A073930

Numbers \(n\) such that \(n\) = sum of the anti-divisors of \(n\).

5, 8, 41, 56, 946, 5186, 6874, 8104, 17386, 27024, 84026, 167786, 2667584, 4921776, 27914146, 505235234, 3238952914, 73600829714, 455879783074, 528080296234, 673223621664, 4054397778846, 4437083907194, 4869434608274, 6904301600914, 7738291969456

This extends the list of such numbers shown in Figure 1.

Anti-multiperfect numbers are integers that equals some product of the sum of its anti-divisors. See Figure 2.
Figure 2
An anti-amicable pair of numbers are two or more numbers such that the sum of the anti-divisors of one equals another, and then the sum of the anti-divisors of this number equal the original number. The idea here can be extended to form chains of anti-amicable numbers - these can be referred to as anti-sociable numbers. See Figure 3.
Figure 4
Some numbers have a large number of anti-divisors. For example, 2139637 has 155 anti-divisors:
2, 3, 5, 7, 9, 11, 13, 15, 19, 21, 25, 33, 34, 35, 39, 45, 53, 55, 57, 63, 65, 75, 77, 86, 91, 95, 99, 105, 117, 133, 143, 165, 171, 175, 195, 209, 225, 231, 247, 263, 273, 275, 285, 307, 315, 325, 385, 399, 429, 455, 475, 495, 525, 585, 627, 665, 693, 715, 741, 819, 825, 855, 975, 1001, 1045, 1155, 1197, 1235, 1287, 1365, 1425, 1462, 1463, 1575, 1729, 1881, 1925, 1995, 2145, 2223, 2275, 2475, 2717, 2925, 3003, 3135, 3325, 3465, 3575, 3705, 4095, 4275, 4389, 5005, 5187, 5225, 5775, 5854, 5985, 6175, 6435, 6825, 7315, 8151, 8645, 9009, 9405, 9975, 10725, 11115, 13167, 13585, 13939, 15015, 15561, 15675, 16271, 17325, 18525, 19019, 20475, 21945, 24453, 25025, 25935, 29925, 32175, 36575, 40755, 43225, 45045, 47025, 55575, 57057, 65835, 67925, 75075, 77805, 80741, 95095, 99518, 109725, 122265, 129675, 171171, 203775, 225225, 251722, 285285, 329175, 389025, 475475, 611325, 855855, 1426425

Figure 5 shows a table of the first few maximally anti-divisible (MAD) natural numbers:

Figure 5

As can be seen the list in Figure 5 is missing some NADs (Number of Anti-Divisors). The full list can be found is OEIS

A066464

Least number \(k\) such that \(k\) has \(n\) anti-divisors.

The full list runs like this where first column is \(n\) and second column is \(k\).

0        1
1        3
2        5
3        7
4        13
5        17
6        32
7        38
8        85
9        67
10    162
11    137
12    338
13    203
14    760
15    247
16    1225
17    472
18    578
19    682
20    1012
21    787
22    9112
23    1463
24    2048
25    2047
26    2888
27    2363
28    5513
29    3465
30    5512
31    6682
32    8978
33    5197
34    17672
35    5198
36    71442
37    9653
38    29768
39    8662
40    40898
41    13513
42    81608
43    15593
44    131072
45    35437
46    49612
47    26163
48    74498
49    22522
50    37538

A MAD run is a series of MAD numbers such that the NAD (Number of Anti-Divisors) value increases by 2. Figure 5 shows that there is an opening MAD run from 1 to 29 and after this, they become rarer, and almost disappear. Notably 293,295,297,299,301 forms a MAD run of 5.
MAD twins are consecutive MAD numbers. Figure 6 shows these:

Figure 6

Most of this content has come from an additional link to the Internet Archive and forms a sort of parallel anti-matter universe to the one that most people are familiar. There are many interesting sequences in the OEIS, some of which are listed above. Here are some more:

A178029

Numbers whose sum of divisors equals the sum of their anti-divisors.