mathematics

Superior highly composite number

Superior highly composite 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 Superior highly composite number rather than just read about it. In short: In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power.

Key takeaways

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

Reference excerpt

In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power. For any possible exponent, whichever integer has the greatest ratio is a superior highly composite number. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer. The first ten superior highly composite numbers and their factorization are listed.

For a superior highly composite number n there exists a positive real number ε > 0 such that for all natural numbers k > 1 we have d ( n ) n ε ≥ d ( k ) k ε {\displaystyle {\frac {d(n)}{n^{\varepsilon }}}\geq {\frac {d(k)}{k^{\varepsilon }}}} where d(n), the divisor function, denotes the number of divisors of n. The term was coined by Ramanujan (1915). For example, the number with the most divisors per square root of the number itself is 12; this can be demonstrated using some highly composites near 12.

2 2 0.5 ≈ 1.414 , 3 4 0.5 = 1.5 , 4 6 0.5 ≈ 1.633 , 6 12 0.5 ≈ 1.732 , 8 24 0.5 ≈ 1.633 , 12 60 0.5 ≈ 1.549 {\displaystyle {\frac {2}{2^{0.5}}}\approx 1.414,{\frac {3}{4^{0.5}}}=1.5,{\frac {4}{6^{0.5}}}\approx 1.633,{\frac {6}{12^{0.5}}}\approx 1.732,{\frac {8}{24^{0.5}}}\approx 1.633,{\frac {12}{60^{0.5}}}\approx 1.549}

120 is another superior highly composite number because it has the highest ratio of divisors to itself raised to the 0.4 power.

9 36 0.4 ≈ 2.146 , 10 48 0.4 ≈ 2.126 , 12 60 0.4 ≈ 2.333 , 16 120 0.4 ≈ 2.357 , 18 180 0.4 ≈ 2.255 , 20 240 0.4 ≈ 2.233 , 24 360 0.4 ≈ 2.279 {\displaystyle {\frac {9}{36^{0.4}}}\approx 2.146,{\frac {10}{48^{0.4}}}\approx 2.126,{\frac {12}{60^{0.4}}}\approx 2.333,{\frac {16}{120^{0.4}}}\approx 2.357,{\frac {18}{180^{0.4}}}\approx 2.255,{\frac {20}{240^{0.4}}}\approx 2.233,{\frac {24}{360^{0.4}}}\approx 2.279}

The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A002201 in the OEIS) are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither set, however, is a subset of the other.

Properties

All superior highly composite numbers are highly composite. This is easy to prove: if there is some number k that has the same number of divisors as n but is less than n itself (i.e. d ( k ) = d ( n ) {\displaystyle d(k)=d(n)} , but k < n {\displaystyle k<n} ), then d ( k ) k ε > d ( n ) n ε {\displaystyle {\frac {d(k)}{k^{\varepsilon }}}>{\frac {d(n)}{n^{\varepsilon }}}} for all positive ε, so if a number "n" is not highly composite, it cannot be superior highly composite. Furthermore, all superior highly numbers have the greatest possible number of divisors for the number of (not necessarily distinct) prime factors it has. An effective construction of the set of all superior highly composite numbers is given by the following monotonic mapping from the positive real numbers. Let

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Illustrations

Worked examples

Example 1 — a first encounter with Superior highly composite number

Start with the simplest possible case. Write down what Superior highly composite 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 Superior highly composite 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 Superior highly composite 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 Superior highly composite number

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

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

Frequently asked questions

What is Superior highly composite number in simple terms?

Why does Superior highly composite 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 Superior highly composite 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 Superior highly composite number.