mathematics

Primitive abundant number

Primitive abundant 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 Primitive abundant number rather than just read about it. In short: In mathematics, a primitive abundant number is an abundant number whose proper divisors are all deficient numbers. For example, 20 is a primitive abundant number because: The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number.

Key takeaways

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

Reference excerpt

In mathematics, a primitive abundant number is an abundant number whose proper divisors are all deficient numbers. For example, 20 is a primitive abundant number because:

The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number. The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8 respectively, so each of these numbers is a deficient number. The first few primitive abundant numbers are:

20, 70, 88, 104, 272, 304, 368, 464, 550, 572 ... (sequence A071395 in the OEIS) The smallest odd primitive abundant number is 945. A variant definition is abundant numbers having no abundant proper divisor, which can also include divisors that are perfect numbers. It starts:

12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114 ... (sequence A091191 in the OEIS)

Properties Every multiple of a primitive abundant number is an abundant number. Every abundant number is a multiple of a primitive abundant number or a multiple of a perfect number. Every primitive abundant number is either a primitive semiperfect number or a weird number. There are an infinite number of primitive abundant numbers. The number of primitive abundant numbers less than or equal to n is o ( n log 2 ⁡ ( n ) ) {\displaystyle o\left({\frac {n}{\log ^{2}(n)}}\right)\,} .

References

Illustrations

Worked examples

Example 1 — a first encounter with Primitive abundant number

Start with the simplest possible case. Write down what Primitive abundant 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 Primitive abundant 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 Primitive abundant 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 Primitive abundant number

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

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

Frequently asked questions

What is Primitive abundant number in simple terms?

In mathematics, a primitive abundant number is an abundant number whose proper divisors are all deficient numbers. For example, 20 is a primitive abundant number because: The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number.

Why does Primitive abundant 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 Primitive abundant 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 Primitive abundant number.