Quasiperfect

In number theory, a quasiperfect number is a natural number n for which sigma(n) = 2n + 1, where sigma(n) is the sum of all positive divisors of n. The concept parallels that of perfect numbers, which satisfy sigma(n) = 2n, but with the additional 1 on the right-hand side.

As of now, no example is known, and the existence of a quasiperfect number remains an open problem. The question has attracted interest as a test case for understanding divisor sums and the structure of integers with near-miss properties to perfect numbers.

Several constraints have been established for quasiperfect numbers. If such a number exists, it must be odd and squareful (every prime factor occurs with exponent at least 2). In addition, researchers have derived various congruence and structural restrictions on the prime factorization, indicating that a hypothetical quasiperfect number would be highly composite and large. Despite these results, no definitive example or universal nonexistence proof is known.

Quasiperfect numbers occupy a niche in the study of divisor-sum phenomena, illustrating how close other natural numbers can come to the ideal of perfectness. They are a topic of ongoing mathematical curiosity, with investigations focusing on what their existence would imply about number factorization and arithmetic functions.