hyperperfect number

Etymology
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Noun

 * 1)  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.
 * 2) * 1966, American Mathematical Society Translations, page 258,
 * the asymptotic density of all hyperperfect numbers, that is, numbers m for which m | σ(m), is equal to zero.

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

Making the relationship with 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 k. A 1-hyperperfect number (or unitary hyperperfect number) is a perfect number.