Aliquot_Sum.md

Aliquot Sum

The aliquot sum $s(n)$ of a positive integer $n$ is the sum of all proper divisors of $n$, that is, all divisors of $n$ other than the number $n$ itself. That is:

$$ s(n) = \sum_{d | n, d \neq n} {d} $$

So, for example, the aliquot sum of the number $15$ is $(1 + 3 + 5) = 9$

Aliquot sum is a very useful property in Number Theory, and can be used for defining:

• Prime Numbers
• Deficient Numbers
• Abundant Numbers
• Perfect Numbers
• Amicable Numbers
• Untouchable Numbers
• Aliquot Sequence of a number
• Quasiperfect & Almost Perfect Numbers
• Sociable Numbers

Facts about Aliquot Sum

• 1 is the only number whose aliquot sum is 0
• The aliquot sums of perfect numbers is equal to the numbers itself
• For a semiprime number of the form $pq$, the aliquot sum is $p + q + 1$
• The Aliquot sum function was one of favorite topics of investigation for the world famous Mathematician, Paul Erdős

Approach on finding the Aliquot sum

Step 1: Obtain the proper divisors of the number

We loop through all the numbers from $1$ to $[\frac{n} 2]$ and check if they divide $n$, which if they do we add them as a proper divisor.

The reason we take the upper bound as $[\frac{n} 2]$ is that, the largest possible proper divisor of an even number is $\frac{n} 2 $, and if the number is odd, then its largest proper divisor is less than $[\frac{n} 2]$, hence making it a foolproof upper bound which is computationally less intensive than looping from $1$ to $n$.

Step 2: Add the proper divisors of the number

The sum which we obtain is the aliquot sum of the number

Implementations

• C#
• C++
• Go
• Java
• JavaScript
• Nim
• Python
• Ruby
• Rust
• TypeScript

Sources

• Wikipedia
• GeeksForGeeks