Hyperperfect number

In number theory, a $k$-hyperperfect number is a natural number $n$ for which the equality $$n = 1+k(\sigma(n)-n-1)$$ 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 $$p = \tfrac{3k+1}{2}$$ and $$q = 3k+4$$ are prime numbers, then $k$ 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 $p^2q$ is an integer such that $$k(p+q) = pq-1,$$ then $k$ 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 $pq$ and corresponding values of $k$ for which $k$ is $k$-hyperperfect:

There are some Even Numbers which are Hyperperfect for Odd Factors i.e., k * (Sum of Odd Factors except 1 and Itself) + 1 = number. E.g., the first 5 ones include 1300, 271872, 304640, 953344 and 1027584 for k = 3, 349, 353, 837 and 353. All Odd Hyperperfect Numbers are Odd Factor Hyperperfect Numbers as they only have odd factors and have no even factors.

1300 has Factors = 1, 2, 4, 5, 10, 13, 20, 25, 26, 50, 52, 65, 100, 130, 260, 325, 650, 1300

It has Odd Factors except 1 and Itself = 5, 13, 25, 65, 325

Sum of Odd Factors except 1 and Itself = 5 + 13 + 25 + 65 + 325 = 433

1300 - 1 = 1299 and 1299/433 = 3, an Integer

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

Definition (Minoli 2010): For any integer $i$ and for integer $k > 0$, define the $n$-hyperdeficiency (or simply the hyperdeficiency) for the number $k$ as

$$\delta_k(n) = n(k+1) + (k-1) - k\sigma(n)$$

A number $i$ is said to be $n$-hyperdeficient if $$\delta_k(n) > 0.$$

Note that for $k = 1$ one gets $$\delta_1(n) = 2n-\sigma(n),$$ which is the standard traditional definition of deficiency.

Lemma: A number $k$ is $k$-hyperperfect (including $k = 1$) if and only if the $i$-hyperdeficiency of $n$, $$\delta_k(n) = 0.$$ Lemma: A number $k$ is $n$-hyperperfect (including $k = 1$) if and only if for some $n$, $$\delta{k-j}(n) = -\delta_{k+j}(n)$$ for at least one $j > 0$.

Books

 * Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p. 114-134)