mathematics

Unusual number

Unusual 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 Unusual number rather than just read about it. In short: In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than n {\displaystyle {\sqrt {n}}} . A k-smooth number has all its prime factors less than or equal to k, therefore, an unusual number is non- n {\displaystyle {\sqrt {n}}} -smooth.

Key takeaways

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

Reference excerpt

In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than n {\displaystyle {\sqrt {n}}} . A k-smooth number has all its prime factors less than or equal to k, therefore, an unusual number is non- n {\displaystyle {\sqrt {n}}} -smooth. The term "unusual number" was coined by Daniel Greene and Donald Knuth, who also showed that, somewhat confusingly, they are asymptotically more dense than their "usual" counterparts.

Relation to prime numbers All prime numbers are unusual. For any prime p, its multiples less than p2 are unusual, that is p, ... (p − 1)p, which have a density 1/p in the interval (p, p2).

Examples The first few unusual numbers are

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, ... (sequence A064052 in the OEIS) The first few non-prime (composite) unusual numbers are

6, 10, 14, 15, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 42, 44, 46, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 99, 102, ... (sequence A063763 in the OEIS)

Distribution If we denote the number of unusual numbers less than or equal to n by u(n) then u(n) behaves as follows:

Richard Schroeppel stated in the HAKMEM (1972), Item #29 that the asymptotic probability that a randomly chosen number is unusual is ln(2). In other words:

lim n → ∞ u ( n ) n = ln ⁡ ( 2 ) = 0.693147 … . {\displaystyle \lim _{n\rightarrow \infty }{\frac {u(n)}{n}}=\ln(2)=0.693147\dots \,.}

References

External links

Weisstein, Eric W. "Rough Number". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Unusual number

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

In research: Unusual 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 Unusual 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: Unusual 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 Unusual 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Unusual number” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Unusual number in 20 minutes

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

Frequently asked questions

What is Unusual number in simple terms?

Why does Unusual 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 Unusual 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 Unusual number.