Hyperperfect number

English

Etymology

From Template:Prefix.

Noun

1. Template:Label Any natural number n for which, for some positive integer k, n = 1 + k(σ(n) - n - 1), where σ(n) is the sum of the positive divisors of n.
   • 1966, American Mathematical Society Translations, page 258,

     Template:...the asymptotic density of all hyperperfect numbers, that is, numbers m for which m | σ(m), is equal to zero.

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Usage notes

Note that hyperperfect numbers are more numerous than perfect numbers (since all perfect numbers are hyperperfect).

Making the relationship with Template:M slightly clearer, the defining equation is sometimes rendered as $n=1+k\sum _i{d}_i$, where the terms $d_i$ are the proper divisors of n (in this context, excluding both 1 and n). n is also said to be a kTemplate:M. A 1-hyperperfect number (or unitary hyperperfect number) is a perfect number.

Derived terms

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Related terms

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