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" − "p" + 1 is prime, "n" = "p""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 A number "n" is said to be k-hyperdeficient if δ("n") > 0. Note that for "k"=1 one gets δ("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("n") = 0. Lemma: A number "n" is k-hyperperfect (including "k"=1) if and only if for  some "k", δ("n") = -δ("n") for at least one "j" > 0.