Anti-Perfect Numbers

The topic of anti-perfect numbers has everything to do with anti-divisors with parallels to perfect numbers and divisors. Let's recall that a perfect number is a number whose sum of proper divisors equals the number. The first perfect number is 6 and it is perfect because its proper divisors of 1, 2 and 3 add to 6. Similarly, an anti-perfect number is a number whose sum of anti-divisors equals the number. The first anti-perfect number is 5 because its anti-divisors of 2 and 3 add to 5.

Now I've written about anti-divisors before in two posts, one titled Anti-divisors on February 26th 2015 and another titled More on Anti-divisors on February 28th 2021. Today, my diurnal age is 27024 and this number has anti-divisors of 7, 32, 49, 96, 1103, 7721 and 18016. The sum of these anti-divisors is 27024 and so 27024 is an anti-perfect number. Like the perfect numbers, these anti-perfect numbers are few and far between. They constitute OEIS A073930:

A073930

Numbers that are equal to the sum of their anti-divisors.

Up to one million, the members of this sequence are 5, 8, 41, 56, 946, 5186, 6874, 8104, 17386, 27024, 84026, 167786, 2667584, 4921776. Thus 27024 is the last anti-perfect number that I'll encounter in my lifetime, hence my celebration of it. The perfect numbers are even sparcer: 6, 28, 496, 8128, 33550336, ... and so I've long ago also reached the last perfect number that I'll encounter in my lifetime.

Let's display the result again:$$ \underbrace{7+32+49+96+1103+7721+18016}_{\text{anti-divisors of }27024} =27024$$It can be noted that the largest anti-divisor, 18016, is exactly 2/3 of 27024 and the largest anti-divisor of any number is always either exactly 2/3 of that number or very close to it. Figure 1 shows a table of anti-perfect numbers and their anti-divisors.