Aliquot sequence

An aliquot sequence is a sequence of numbers where each number is the sum of the proper divisors of the previous number. Proper divisors are all the divisors of a number except the number itself.

The word "aliquot" comes from Latin. It means "a part of the whole."

1. Start with a number, called the starting number.
2. Find the sum of the proper divisors of that number.
3. Use this sum as the next number in the sequence.
4. Repeat the process.

For example, if the starting number is 12, the proper divisors of 12 are 1, 2, 3, 4, and 6.

• Their sum is 1+2+3+4+6=16, so the next number in the sequence is 16.
• The proper divisors of 16 are 1, 2, 4, and 8.
• Their sum is 1+2+4+8=15, so the next number is 15.
• The proper divisors of 15 are 1, 3, and 5.
• Their sum is 1+3+5=9, and so on.

1. Terminating sequence: If the sequence reaches 1, it stops because the only proper divisor of 1 is 0.
   • Example: Starting with 6, the sequence is 6 → 6 → 6 (this is also called a perfect number sequence).
2. Perfect numbers: If the starting number is a perfect number, the sequence stays the same. For example, starting with 28, the sequence is 28 → 28 → 28.
3. Amicable numbers: If the sequence alternates between two numbers, those two numbers are called amicable numbers. For example, starting with 220, the sequence is 220 → 284 → 220 → 284.
4. Sociable numbers: If the sequence cycles through more than two numbers, those numbers are called sociable numbers.
5. Non-terminating sequence: For some starting numbers, the sequence continues indefinitely or grows very large. The smallest number with this property is 276, since its aliquot sum is 396. Additionally, 306 is also the start of the same infinite aliquot sequence since its aliquot sum is also 396.

• Aliquot sequences are studied in number theory.
• They help mathematicians understand the relationships between numbers and their divisors.
• Perfect, amicable, and sociable numbers are special types of numbers studied using aliquot sequences.