Hyperperfect number

In number theory, a k-hyperperfect number is a natural number n for which the equality ${\displaystyle n=1+k(\sigma (n)-n-1)}$ 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.^[1]

The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... (sequence A034897 in the OEIS), with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... (sequence A034898 in the OEIS). The first few k-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... (sequence A007592 in the OEIS).

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 ${\displaystyle p={\tfrac {3k+1}{2}}}$ and ${\displaystyle q=3k+4}$ are prime numbers, then ⁠${\displaystyle p^{2}q}$⁠ 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 ${\displaystyle k(p+q)=pq-1,}$ then pq is k-hyperperfect.

It is also possible to show that if k > 0 and ${\displaystyle p=k+1}$ is prime, then for all i > 1 such that ${\displaystyle q=p^{i}-p+1}$ is prime, ${\displaystyle n=p^{i-1}q}$ is k-hyperperfect. The following table lists known values of k and corresponding values of i for which n is k-hyperperfect:

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

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

$${\displaystyle \delta _{k}(n)=n(k+1)+(k-1)-k\sigma (n)}$$

A number n is said to be k-hyperdeficient if ${\displaystyle \delta _{k}(n)>0.}$

Note that for k = 1 one gets ${\displaystyle \delta _{1}(n)=2n-\sigma (n),}$ 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, ${\displaystyle \delta _{k}(n)=0.}$

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

• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. p. 114. ISBN 1-4020-4215-9. Zbl 1151.11300.

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

• MathWorld: Hyperperfect number
• A long list of hyperperfect numbers under Data