Aliquot sum

https://en.wikipedia.org/wiki/Aliquot_sum

• 1 is the only number whose aliquot sum is 0

• A number is prime iff its aliquot sum is 1

• The aliquot sum of perfect number n = n

• The aliquot sum of deficient number n < n

• The aliquot sum of abundant number n > n

In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n.

Proper divisors of n are all divisors of n other than n itself.

The aliquot sum can be used to characterize the prime numbers, perfect numbers, deficient numbers, abundant numbers and untouchable numbers. Also, to define the aliquot sequence of a number.

For example, the proper divisors of 28 are {1,2,4,7,14}, so the aliquot sum of 28 is 28, making it the perfect number.

The sequence A001065 in the OEIS is made of values of s(n) for ℕᐩ:

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, ...

meaning the aliquot sum of 1 is s(1)=0, s(2)=1, s(3)=1, s(4)=3

n	s(n)	D(n)
1	0	∅
2	1	{1}
3	1	{1}
4	3	{1,2}
5	1	{1}
6	6	{1,2,3}
7	1	{1}
8	7	{1,2,4}
9	4	{1,3}
10	8	{1,2,5}