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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.

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).

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 ${\displaystyle p={\tfrac {3k+1}{2}}}$ 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:

Hyperdeficiency

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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.

References

1. Weisstein, Eric W. "Hyperperfect Number". mathworld.wolfram.com. Retrieved 2020-08-10.

• 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.

Further reading

Articles

• Minoli, Daniel; Bear, Robert (Fall 1975), "Hyperperfect numbers", Pi Mu Epsilon Journal, 6 (3): 153–157.
• Minoli, Daniel (Dec 1978), "Sufficient forms for generalized perfect numbers", Annales de la Faculté des Sciences UNAZA, 4 (2): 277–302.
• Minoli, Daniel (Feb 1981), "Structural issues for hyperperfect numbers", Fibonacci Quarterly, 19 (1): 6–14, doi:10.1080/00150517.1981.12430116.
• Minoli, Daniel (April 1980), "Issues in non-linear hyperperfect numbers", Mathematics of Computation, 34 (150): 639–645, doi:10.2307/2006107, JSTOR 2006107.
• Minoli, Daniel (October 1980), "New results for hyperperfect numbers", Abstracts of the American Mathematical Society, 1 (6): 561.
• Minoli, Daniel; Nakamine, W. (1980). "Mersenne numbers rooted on 3 for number theoretic transforms". ICASSP '80. IEEE International Conference on Acoustics, Speech, and Signal Processing. Vol. 5. pp. 243–247. doi:10.1109/ICASSP.1980.1170906..
• McCranie, Judson S. (2000), "A study of hyperperfect numbers", Journal of Integer Sequences, 3: 13, Bibcode:2000JIntS...3...13M, archived from the original on 2004-04-05.
• te Riele, Herman J.J. (1981), "Hyperperfect numbers with three different prime factors", Math. Comp., 36 (153): 297–298, doi:10.1090/s0025-5718-1981-0595066-9, MR 0595066, Zbl 0452.10005.
• te Riele, Herman J.J. (1984), "Rules for constructing hyperperfect numbers", Fibonacci Q., 22: 50–60, doi:10.1080/00150517.1984.12429920, Zbl 0531.10005.

Books

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

External links

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

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect

• Divisor function
• Integer sequences
• Perfect numbers