Add Aliquot Sum Explanation (TheAlgorithms#188)

SpiderMath · web-flow · commit 0c2e2e6ed5d0 · 2022-10-17T12:54:21.000+02:00
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+# 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*](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Paul_Erd%C5%91s) 
+
+## 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#](https://github.com/TheAlgorithms/C-Sharp/blob/master/Algorithms/Numeric/AliquotSumCalculator.cs)
+- [Java](https://github.com/TheAlgorithms/Java/blob/master/src/main/java/com/thealgorithms/maths/AliquotSum.java)
+- [JavaScript](https://github.com/TheAlgorithms/JavaScript/blob/master/Maths/AliquotSum.js)
+- [Python](https://github.com/TheAlgorithms/Python/blob/master/maths/aliquot_sum.py)
+- [Ruby](https://github.com/TheAlgorithms/Ruby/blob/master/maths/aliquot_sum.rb)
+
+## Sources
+- [Wikipedia](https://en.wikipedia.org/wiki/Aliquot_sum)
+- [GeeksForGeeks](https://www.geeksforgeeks.org/aliquot-sum/)
