Aliquot sum

Pick a number. Find its smaller parts. These parts must fit into it evenly. Do not use the number itself. Add those smaller parts together. This total is called an aliquot sum.

Some numbers are special. Prime numbers always have a sum of one. One is the only number with a sum of zero.

Some numbers are called perfect. Their sum is equal to the number. Others are called abundant or deficient. This depends on the sum.

Some numbers are untouchable. No other number has them as a sum. Two and five are untouchable.

Math can show us these patterns. It is a way to group numbers.

Think about a number. Find all the smaller numbers that fit into it evenly. Do not use the number itself. Add these smaller parts together. This total is called an aliquot sum.

Let us look at the number 12. Its parts are 1, 2, 3, 4, and 6. When we add them, we get 16. This is the aliquot sum for 12.

Many types of numbers use this sum to show their kind. Prime numbers always have a sum of 1. The number 1 is the only one with a sum of 0.

Some numbers are perfect. This means their sum equals the number itself. Other numbers are abundant if their sum is larger. Some are deficient if their sum is smaller.

There are also untouchable numbers. No other number has them as a sum. Abu Mansur al-Baghdadi studied these long ago. He saw that 2 and 5 are untouchable. Paul Erdős proved there are infinite untouchable numbers. Some people think 5 is the only odd one. We do not know for sure yet.

If you use the sum to find a new sum, you make a sequence. Some numbers form a loop. We call these sociable numbers. Amicable numbers are a special kind of loop with two steps.

Imagine you have a number. You want to find all the smaller numbers that fit into it perfectly. These are called divisors. To find the aliquot sum, you find all these divisors. You must leave out the number itself. Then, you add all those smaller parts together. This total is the aliquot sum. It is a special way to study how numbers work.

Let us try an example with the number 12. The numbers that fit into 12 are 1, 2, 3, 4, 6, and 12. We do not use 12 for this sum. We only add 1, 2, 3, 4, and 6. These five parts add up to 16. So, the aliquot sum of 12 is 16. This simple step helps us group numbers into different families.

People have studied these sums for a very long time. A man named Abu Mansur al-Baghdadi studied them around 1000 AD. He noticed that 2 and 5 are untouchable numbers. An untouchable number is a sum that no other number can make. Later, a mathematician named Paul Erdős proved these numbers go on forever. He also loved studying the aliquot sum function.

We can use these sums to name many kinds of numbers. A prime number always has an aliquot sum of 1. The number 1 is the only one with a sum of 0. Perfect numbers have a sum equal to themselves. Abundant numbers have a sum that is larger. Deficient numbers have a sum that is smaller. There are also almost perfect numbers, like the powers of 2.

Sometimes, you can use a sum to find a new sum. This creates a chain called an aliquot sequence. Some numbers create a loop in their chain. We call these sociable numbers. If the loop has only two steps, they are called amicable numbers. Mathematicians still wonder about these chains. They do not know if every chain ends in a loop or a prime number.

In the field of number theory, mathematicians study the properties of integers. One specific tool used for this study is the aliquot sum. The aliquot sum of a positive integer is the sum of all its proper divisors. A proper divisor is any positive divisor of a number except for the number itself. This mathematical function allows researchers to categorize numbers into distinct families based on their internal structure. By looking at these sums, we can understand the fundamental relationships between different integers.

To calculate an aliquot sum, you must follow a specific sequence of steps. First, identify every positive integer that divides into your chosen number without leaving a remainder. Next, you must exclude the original number from this list of divisors. These remaining values are your proper divisors. Finally, you add all these proper divisors together to reach the total. For example, consider the number 12. Its divisors are 1, 2, 3, 4, 6, and 12. To find the aliquot sum, you ignore 12 and add 1, 2, 3, 4, and 6. This calculation results in 16.

Mathematicians use these sums to define several important classes of numbers. The number 1 is unique because its aliquot sum is 0. A prime number is easily identified because its aliquot sum is always exactly 1. We can also group numbers by comparing their aliquot sum to the original value. If the sum is less than the number, it is called a deficient number. If the sum is greater than the number, it is an abundant number. If the sum is exactly equal to the number, it is a perfect number.

There are even more specialized categories within this system. Quasiperfect numbers are those whose aliquot sums equal the original number plus two. While these are theorized, it is not known if they actually exist. Almost perfect numbers are those where the aliquot sum equals the original number minus one. The powers of 2 are the only known examples of almost perfect numbers. Another category involves untouchable numbers. An untouchable number is a value that is not the aliquot sum of any other number.

The study of these values has a long and fascinating history. Around 1000 AD, Abu Mansur al-Baghdadi observed that the numbers 2 and 5 are untouchable. This early work laid the groundwork for much more complex proofs. Later, the mathematician Paul Erdős investigated these functions deeply. Erdős proved that the set of untouchable numbers is infinite. He also considered the aliquot sum function to be one of his favorite subjects of investigation.

One of the most interesting ways to use this function is through iteration. Iteration means taking the result of a calculation and using it as the next input. This process creates what is known as an aliquot sequence. Some sequences eventually enter a repeating loop. Numbers that belong to such a periodic sequence are called sociable numbers. If the sequence has a period of exactly two, the numbers are called amicable numbers.

Despite centuries of study, many questions about these sequences remain unanswered. Mathematicians do not yet know if every aliquot sequence will eventually end in a specific way. Some sequences might end with a prime number or a perfect number. Others might end by falling into a periodic sequence of sociable numbers. There is also an unproven conjecture regarding untouchable numbers. It is suggested that 5 is the only odd untouchable number. This idea connects to Goldbach's conjecture and the properties of semiprime numbers.