mathematics

Semiperfect number

Semiperfect number is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Semiperfect number rather than just read about it. In short: In number theory, a semiperfect number or pseudoperfect number is a natural number n equal to the sum of all or some of its proper divisors. A semiperfect number equal to the sum of all its proper divisors is a perfect number.

Key takeaways

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

Reference excerpt

In number theory, a semiperfect number or pseudoperfect number is a natural number n equal to the sum of all or some of its proper divisors. A semiperfect number equal to the sum of all its proper divisors is a perfect number. The first few semiperfect numbers are: 6, 12, 18, 20, 24, 28, 30, 36, 40, ... (sequence A005835 in the OEIS)

Properties Every multiple of a semiperfect number is semiperfect. A semiperfect number not divisible by any smaller semiperfect number is called primitive. Every number of the form 2mp for a natural number m and an odd prime number p such that p < 2m+1 is also semiperfect. In particular, every number of the form 2m(2m+1 − 1) is semiperfect, and is indeed perfect if 2m+1 − 1 is a Mersenne prime. The smallest odd semiperfect number is 945. A semiperfect number is necessarily either perfect or abundant. An abundant number that is not semiperfect is called a weird number. Except for 2, all primary pseudoperfect numbers are semiperfect. Every practical number that is not a power of two is semiperfect. The natural density of the set of semiperfect numbers exists.

Primitive semiperfect numbers A primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a semiperfect number that has no semiperfect proper divisor. The first few primitive semiperfect numbers are 6, 20, 28, 88, 104, 272, 304, 350, ... (sequence A006036 in the OEIS) There are infinitely many such numbers. All numbers of the form 2mp, with p a prime between 2m and 2m+1, are primitive semiperfect, but not all primitive semiperfect numbers follow this form; for example, 770. There are infinitely many odd primitive semiperfect numbers, the smallest being 945. There are infinitely many primitive semiperfect numbers that are not harmonic divisor numbers. Every semiperfect number is a multiple of a primitive semiperfect number.

See also Hemiperfect number Erdős–Nicolas number

Notes

References Friedman, Charles N. (1993). "Sums of divisors and Egyptian fractions". Journal of Number Theory. 44 (3): 328–339. doi:10.1006/jnth.1993.1057. MR 1233293. Zbl 0781.11015. Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-20860-7. OCLC 54611248. Zbl 1058.11001. Section B2. Sierpiński, Wacław (1965). "Sur les nombres pseudoparfaits". Mat. Vesn. Nouvelle Série (in French). 2 (17): 212–213. MR 0199147. Zbl 0161.04402. Zachariou, Andreas; Zachariou, Eleni (1972). "Perfect, semiperfect and Ore numbers". Bull. Soc. Math. Grèce. Nouvelle Série. 13: 12–22. MR 0360455. Zbl 0266.10012.

External links Weisstein, Eric W. "Pseudoperfect Number". MathWorld. Weisstein, Eric W. "Primitive Pseudoperfect Number". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Semiperfect number

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

In research: Semiperfect 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 Semiperfect 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: Semiperfect number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, Perfect numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Semiperfect 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.

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How to study Semiperfect number in 20 minutes

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

Frequently asked questions

What is Semiperfect number in simple terms?

Why does Semiperfect 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 Semiperfect 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 Semiperfect number.