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Hyperperfect number - mklabgr - 08-05-2026
Hyperperfect number
Summary
A hyperperfect number is a concept in number theory that generalizes the idea of a perfect number. A natural number $n$ is called $k$-hyperperfect if it satisfies the equation $n = 1 + k(\sigma(n)-n-1)$, where $\sigma(n)$ is the sum of all positive divisors of $n$.
In this formula, $k$ is a positive integer, and when $k=1$, the definition reduces to that of a perfect number, meaning that the number equals the sum of its proper divisors. Hyperperfect numbers include all perfect numbers but also many additional examples; for instance, the first non-perfect hyperperfect numbers include $21$, $2133$, and $19521$.
Mathematicians study these numbers to understand deeper relationships between divisors, divisor sums, and special structures in integers, although many questions about their distribution and properties remain open.
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