Hyperperfect number

In mathematics, a "k"-hyperperfect number is a natural number "n" for which the equality "n" = 1 + "k"("σ"("n") − "n" − 1) holds, where "σ"("n") is the divisor function (i.e., the sum of all positive divisors of "n"). A hyperperfect number is a "k"-hyperperfect number for some integer "k". Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.
The first few numbers in the sequence of "k"-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... , with the corresponding values of "k" being 1, 2, 1, 6, 3, 1, 12, ... . The first few "k"-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... .
List of hyperperfect numbers.
The following table lists the first few "k"-hyperperfect numbers for some values of "k", together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of "k"-hyperperfect numbers:
It can be shown that if "k" > 1 is an odd integer and "p" = (3"k" + 1) / 2 and "q" = 3"k" + 4 are prime numbers, then "p"²"q" is "k"-hyperperfect; Judson S. McCranie has conjectured in 2000 that all "k"-hyperperfect numbers for odd "k" > 1 are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if "p" ≠ "q" are odd primes and "k" is an integer such that "k"("p" + "q") = "pq" - 1, then "pq" is "k"-hyperperfect.
It is also possible to show that if "k" > 0 and "p" = "k" + 1 is prime, then for all "i" > 1 such that "q" = "p""i" − "p" + 1 is prime, "n" = "p""i" − 1"q" is "k"-hyperperfect. The following table lists known values of "k" and corresponding values of "i" for which "n" is "k"-hyperperfect:
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", formula_1, define the k-hyperdeficiency (or simply the hyperdeficiency) for the number "n" as
 δk(n) = n(k+1) +(k-1) – kσ(n)
A number "n" is said to be k-hyperdeficient if δ"k"("n") > 0.
Note that for "k"=1 one gets δ1("n")= 2"n"–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.