Hyperperfect Number - Hyperdeficiency

The newly-introduced mathematical concept of hyperdeficiency is related to the hyperperfect numbers.

Definition (Minoli 2010): For any integer n and for integer k, -∞k-hyperdeficiency (or simply the hyperdeficiency) for the number n as

A number n is said to be k-hyperdeficient if δ_k(n) > 0.

Note that for k=1 one gets δ₁(n)= 2n–σ(n), which is the standard traditional definition of deficiency.

Lemma: A number n is k-hyperperfect (including k=1) if and only if the k-hyperdeficiency of n, δ_k(n) = 0.

Lemma: A number n is k-hyperperfect (including k=1) if and only if for some k, δ_k-j(n) = -δ_k+j(n) for at least one j > 0.