Quasiperfect number

Key takeaways

Quasiperfect 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 Quasiperfect number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Quasiperfect number from memory before moving on to harder problems.

Reference excerpt

In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the sum-of-divisors function σ ( n ) {\displaystyle \sigma (n)} ) is equal to 2 n + 1 {\displaystyle 2n+1} . Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far. The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1).

Theorems If a quasiperfect number exists, it must be an odd square number greater than 1035 and have at least seven distinct prime factors.

Related For a perfect number n the sum of all its divisors is equal to 2 n {\displaystyle 2n} . For an almost perfect number n the sum of all its divisors is equal to 2 n − 1 {\displaystyle 2n-1} . Numbers n whose sum of factors equals 2 n + 2 {\displaystyle 2n+2} are known to exist. They are of form 2 n − 1 × ( 2 n − 3 ) {\displaystyle 2^{n-1}\times (2^{n}-3)} where 2 n − 3 {\displaystyle 2^{n}-3} is a prime. The only exception known so far is 650 = 2 × 5 2 × 13 {\displaystyle 650=2\times 5^{2}\times 13} . They are 20, 104, 464, 650, 1952, 130304, 522752, ... (sequence A088831 in the OEIS). Numbers n whose sum of factors equals 2 n − 2 {\displaystyle 2n-2} are also known to exist. They are of form 2 n − 1 × ( 2 n + 1 ) {\displaystyle 2^{n-1}\times (2^{n}+1)} where 2 n + 1 {\displaystyle 2^{n}+1} is prime. No exceptions are found so far. Because of the five known Fermat primes, there are five such numbers known: 3, 10, 136, 32896 and 2147516416 (sequence A191363 in the OEIS) Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.

Notes

References Brown, E.; Abbott, H.; Aull, C.; Suryanarayana, D. (1973). "Quasiperfect numbers" (PDF). Acta Arith. 22 (4): 439–447. doi:10.4064/aa-22-4-439-447. MR 0316368. Kishore, Masao (1978). "Odd integers N with five distinct prime factors for which 2−10−12 < σ(N)/N < 2+10−12" (PDF). Mathematics of Computation. 32 (141): 303–309. doi:10.2307/2006281. ISSN 0025-5718. JSTOR 2006281. MR 0485658. Zbl 0376.10005. Cohen, Graeme L. (1980). "On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers". J. Austral. Math. Soc. Ser. A. 29 (3): 369–384. doi:10.1017/S1446788700021376. ISSN 0263-6115. MR 0569525. S2CID 120459203. Zbl 0425.10005. James J. Tattersall (1999). Elementary number theory in nine chapters. Cambridge University Press. pp. 147. ISBN 0-521-58531-7. Zbl 0958.11001. Guy, Richard (2004). Unsolved Problems in Number Theory, third edition. Springer-Verlag. p. 74. ISBN 0-387-20860-7. Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 109–110. ISBN 1-4020-4215-9. Zbl 1151.11300.

Worked examples

Example 1 — a first encounter with Quasiperfect number

Start with the simplest possible case. Write down what Quasiperfect 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 Quasiperfect 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 Quasiperfect 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 Quasiperfect number

In research: Quasiperfect 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 Quasiperfect 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: Quasiperfect number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Arithmetic dynamics, Divisor function, Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Quasiperfect 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 Quasiperfect number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Quasiperfect 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 Quasiperfect number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quasiperfect number in simple terms?

In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the sum-of-divisors function σ ( n ) {\displaystyle \sigma (n)} ) is equal to 2 n + 1 {\displaystyle 2n+1} . Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 a…

Why does Quasiperfect 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 Quasiperfect 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 Quasiperfect number.