| The Anti-Divisor |
An interesting ad metric is to sum the anti-divisors of a number. If this happens to be n, then the number is anti-perfect. Most numbers are not anti-perfect.
But, not all of the integers are represented by an anti-divisor sum. Some are represented once, and others more.
Here is a JavaScript program that displays the anti-divisor sums, along with the number of n that produce the same result, and a list of these n.
Note that the data is only accurate to about 333, as this is the highest ad of 499.
A brief summary of the results:
| n | sigma(ads of n) |
|---|---|
| 3 | 2 |
| 4 | 3 |
| 5 | 5 |
| 6 | 4 |
| 7 | 10 |
| 8 | 8 |
| 9 | 8 |
| 10 | 14 |
| 11 | 12 |
| 12 | 13 |
| 13 | 19 |
| 14 | 16 |
| 15 | 18 |
| 16 | 14 |
| 17 | 28 |
| 18 | 28 |
| 19 | 18 |
| 20 | 24 |
Most of the sums in this table are even, and over the range 2-499, we find 468 even ad sums, and only 30 odd ad sums.
The odd ad sums demonstrate a strange pattern - the distance between consecutive odd ad sums are : 1,7,1,11,1,15,1,19,1,23,1,27,1,31,1,35,1,39,1,43,1,47,1,51,1,55,1,59,1
Anti-phiThe anti-phi function is defined as the numbers <n that do not have any anti-divisor as a factor.
| n | anti-phi |
|---|---|
| 2 | 1 |
| 3 | 1 |
| 4 | 2 |
| 5 | 1 |
| 6 | 4 |
| 7 | 1 |
| 8 | 4 |
| 9 | 4 |
| 10 | 3 |
| 11 | 2 |
| 12 | 8 |
| 13 | 3 |
| 14 | 7 |
| 15 | 7 |
| 16 | 9 |
| 17 | 2 |
| 18 | 8 |
| 19 | 5 |
| 20 | 10 |
The first number after the list of ads is the anti-phi value, the second the phi value, and the third is phi+anti-phi.
An interesting feature of this is when phi(n)+anti-phi(n)=n. This occurs for:
2,3,4,5,6,7,8,12,15,27,30,40,44,57,117,128,171,236,399, 408 and 510. And no more to 1000.
Therefore a conjecture is that there are only a finite number of such numbers.
Note that it is possible for anti-phi(i) to divide i-1, e.g. 5 and 15.
Defining anti-divisors by classConsider the ads of 22, i.e. 3,4,5,9 and 15. And consider the ads of 23, i.e. 2,3,5,9 and 15.
These two numbers have severals ads in common, but we can say that the ads belong to different classes. The odd ads of 22 belong to the lower ad class, and the odd ads of 23 belong to the upper ad class.
So a formal definition would be that an even ad has a class value of 0, a lower ad has a class value of -1, and an upper ad has a class value of 1.
| n | ads | ad classes | class sum |
|---|---|---|---|
| 3 | 2. | 0. | 0 |
| 4 | 3. | -1. | -1 |
| 5 | 2.3. | 0.1. | 1 |
| 6 | 4. | 0. | 0 |
| 7 | 2.3.5. | 0.-1.-1. | -2 |
| 8 | 3.5. | 1.1. | 2 |
| 9 | 2.6. | 0.0. | 0 |
| 10 | 3.4.7. | -1.0.-1. | -2 |
| 11 | 2.3.7. | 0.1.1. | 2 |
| 12 | 5.8. | -1.0. | -1 |
| 13 | 2.3.5.9. | 0.-1.1.-1. | -1 |
| 14 | 3.4.9. | 1.0.1. | 2 |
| 15 | 2.6.10. | 0.0.0. | 0 |
| 16 | 3.11. | -1.-1. | -2 |
| 17 | 2.3.5.7.11. | 0.1.-1.-1.1. | 0 |
| 18 | 4.5.7.12. | 0.1.1.0. | 2 |
| 19 | 2.3.13. | 0.-1.-1. | -2 |
| 20 | 3.8.13. | 1.0.1. | 2 |
A class sum is defined as the sum of the ad classes.
Two conjectures:
Please address questions/comments/suggestions to : Jon Perry