Hyperperfect number

Key takeaways

Hyperperfect number belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hyperperfect number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hyperperfect number from memory before moving on to harder problems.

Reference excerpt

In number theory, a k-hyperperfect number is a natural number n for which the equality n = 1 + k ( σ ( n ) − n − 1 ) {\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 p = 3 k + 1 2 {\displaystyle p={\tfrac {3k+1}{2}}} and q = 3 k + 4 {\displaystyle q=3k+4} are prime numbers, then ⁠ p 2 q {\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 k ( p + q ) = p q − 1 , {\displaystyle k(p+q)=pq-1,} then pq is k-hyperperfect. It is also possible to show that if k > 0 and p = k + 1 {\displaystyle p=k+1} is prime, then for all i > 1 such that q = p i − p + 1 {\displaystyle q=p^{i}-p+1} is prime, n = p i − 1 q {\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:

References

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

Worked examples

Example 1 — a first encounter with Hyperperfect number

Start with the simplest possible case. Write down what Hyperperfect number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hyperperfect number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hyperperfect number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hyperperfect number

In research: Hyperperfect number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hyperperfect number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hyperperfect number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Divisor function, Integer sequences, Perfect numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Hyperperfect number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Hyperperfect number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hyperperfect number means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hyperperfect number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hyperperfect number in simple terms?

In number theory, a k-hyperperfect number is a natural number n for which the equality n = 1 + k ( σ ( n ) − n − 1 ) {\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 so…

Why does Hyperperfect number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hyperperfect number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hyperperfect number.