Hyperperfect number

Type of natural 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).

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:

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 ^{[citation needed] [clarification needed]}

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 > 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.
• 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, 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