mathematics

Highly abundant number

Highly 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 Highly abundant number rather than just read about it. In short: In number theory, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Highly abundant numbers and several similar classes of numbers were first introduced by Pillai (1943), and early work on the subject was done by Alaoglu and Erdős (1944).

Key takeaways

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

Reference excerpt

In number theory, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Highly abundant numbers and several similar classes of numbers were first introduced by Pillai (1943), and early work on the subject was done by Alaoglu and Erdős (1944). Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any N is at least proportional to log2 N.

Formal definition and examples Formally, a natural number n is called highly abundant if and only if for all natural numbers m < n,

σ ( n ) > σ ( m ) {\displaystyle \sigma (n)>\sigma (m)}

where σ denotes the sum-of-divisors function. The first few highly abundant numbers are

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... (sequence A002093 in the OEIS). For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ. The only odd highly abundant numbers are 1 and 3.

Relations with other sets of numbers

Although the first eight factorials are highly abundant, not all factorials are highly abundant. For example,

σ(9!) = σ(362880) = 1481040, but there is a smaller number with larger sum of divisors,

σ(360360) = 1572480, so 9! is not highly abundant. Alaoglu and Erdős noted that all superabundant numbers are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant. This question was answered affirmatively by Jean-Louis Nicolas (1969). Despite the terminology, not all highly abundant numbers are abundant numbers. In particular, none of the first seven highly abundant numbers (1, 2, 3, 4, 6, 8, and 10) is abundant. Along with 16, the ninth highly abundant number, these are the only highly abundant numbers that are not abundant. 7200 is the largest powerful number that is also highly abundant: all larger highly abundant numbers have a prime factor that divides them only once. Therefore, 7200 is also the largest highly abundant number with an odd sum of divisors.

Notes

References Alaoglu, L.; Erdős, P. (1944). "On highly composite and similar numbers" (PDF). Transactions of the American Mathematical Society. 56 (3): 448–469. doi:10.2307/1990319. JSTOR 1990319. MR 0011087. Nicolas, Jean-Louis (1969). "Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers"". Bull. Soc. Math. France. 97: 129–191. doi:10.24033/bsmf.1676. MR 0254130. Pillai, S. S. (1943). "Highly abundant numbers". Bull. Calcutta Math. Soc. 35: 141–156. MR 0010560.

Illustrations

Worked examples

Example 1 — a first encounter with Highly abundant number

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

In research: Highly 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 Highly 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: Highly 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 Highly 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 Highly abundant number in 20 minutes

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

Frequently asked questions

What is Highly abundant number in simple terms?

Why does Highly 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 Highly 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 Highly abundant number.