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Hemiperfect number

Hemiperfect 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 Hemiperfect number rather than just read about it. In short: In number theory, a hemiperfect number is a positive integer with a half-integer abundancy index. In other words, σ(n)/n = k/2 for an odd integer k, where σ(n) is the sum-of-divisors function, the sum of all positive divisors of n.

Key takeaways

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

Reference excerpt

In number theory, a hemiperfect number is a positive integer with a half-integer abundancy index. In other words, σ(n)/n = k/2 for an odd integer k, where σ(n) is the sum-of-divisors function, the sum of all positive divisors of n. The first few hemiperfect numbers are:

2, 24, 4320, 4680, 26208, 8910720, 17428320, 20427264, 91963648, 197064960, ... (sequence A159907 in the OEIS)

Example 24 is a hemiperfect number because the sum of the divisors of 24 is

1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = ⁠5/2⁠ × 24. The abundancy index is 5/2 which is a half-integer.

Smallest hemiperfect numbers of abundancy k/2 The following table gives an overview of the smallest hemiperfect numbers of abundancy k/2 for k ≤ 13 (sequence A088912 in the OEIS):

The current best known upper bounds for the smallest numbers of abundancy 15/2 and 17/2 were found by Michel Marcus. The smallest known number of abundancy 15/2 is ≈ 1.274947×1088, and the smallest known number of abundancy 17/2 is ≈ 2.717290×10190. There are no known numbers of abundancy 19/2.

See also Semiperfect number Perfect number Multiply perfect number

References

Worked examples

Example 1 — a first encounter with Hemiperfect number

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

In research
Hemiperfect 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 Hemiperfect 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
Hemiperfect 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 Hemiperfect 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 Hemiperfect number in 20 minutes

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

Frequently asked questions

What is Hemiperfect number in simple terms?

In number theory, a hemiperfect number is a positive integer with a half-integer abundancy index. In other words, σ(n)/n = k/2 for an odd integer k, where σ(n) is the sum-of-divisors function, the sum of all positive divisors of n.

Why does Hemiperfect 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 Hemiperfect 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 Hemiperfect number.

Tags

  • Integer sequences
  • Perfect numbers

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