pseudoperfect

Pseudoperfect is a term used in number theory to describe a composite number that is equal to the sum of all or some of its proper divisors. A proper divisor of a number is any divisor of that number excluding the number itself. For example, the proper divisors of 12 are 1, 2, 3, 4, and 6. The sum of these proper divisors is 1 + 2 + 3 + 4 + 6 = 16. Since 16 is not equal to 12, 12 is not a pseudoperfect number. However, if we consider a subset of these proper divisors, such as 1 + 2 + 3 + 6 = 12, then 12 is a pseudoperfect number.

The term pseudoperfect is synonymous with the term semiperfect. Numbers that are the sum of all of their proper divisors are called perfect numbers, a subset of pseudoperfect numbers. For instance, 6 is a perfect number because its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. Therefore, 6 is also a pseudoperfect number.

The smallest odd pseudoperfect number is 945. The concept of pseudoperfect numbers is related to other classifications of numbers based on the sum of their divisors, such as abundant numbers (where the sum of proper divisors is greater than the number) and deficient numbers (where the sum of proper divisors is less than the number). All perfect numbers are pseudoperfect, but not all pseudoperfect numbers are perfect.