Editing Hyperperfect number (section)

{{short description|Type of natural number}}

In [[number theory]], a '''{{mvar|k}}-hyperperfect number''' is a [[natural number]] {{mvar|n}} for which the equality <math>n = 1+k(\sigma(n)-n-1)</math> holds, where {{math|''σ''(''n'')}} is the [[divisor function]] (i.e., [[Aliquot sum|the sum]] of all positive [[divisor]]s of {{mvar|n}}). A '''hyperperfect number''' is a {{mvar|k}}-hyperperfect number for some integer {{mvar|k}}. Hyperperfect numbers generalize [[perfect number]]s, which are 1-hyperperfect.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Hyperperfect Number|url=https://mathworld.wolfram.com/HyperperfectNumber.html|access-date=2020-08-10|website=mathworld.wolfram.com|language=en}}</ref>

The first few numbers in the sequence of {{mvar|k}}-hyperperfect numbers are {{nowrap|6, 21, 28, 301, 325, 496, 697, ...}} {{OEIS|A034897}}, with the corresponding values of {{mvar|k}} being {{nowrap|1, 2, 1, 6, 3, 1, 12, ...}} {{OEIS|id=A034898}}. The first few {{mvar|k}}-hyperperfect numbers that are not perfect are {{nowrap|21, 301, 325, 697, 1333, ...}} {{OEIS|A007592}}.