mathematics

Refactorable number

Refactorable 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 Refactorable number rather than just read about it. In short: A refactorable number or tau number is an integer n that is divisible by the count of its divisors, or to put it algebraically, n is such that τ ( n ) ∣ n {\displaystyle \tau (n)\mid n} with τ ( n ) = σ 0 ( n ) = ∏ i = 1 n ( e i + 1 ) {\displaystyle \tau (n)=\sigma _{0}(n)=\prod _{i=1}^{n}(e_{i}+1)} for n = ∏ i = 1 n p i e i {\displaystyle n=\prod _{i=1}^{n}p_{i}^{e_{i}}} . The first few refactorable numbers are lis…

Key takeaways

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

Reference excerpt

A refactorable number or tau number is an integer n that is divisible by the count of its divisors, or to put it algebraically, n is such that τ ( n ) ∣ n {\displaystyle \tau (n)\mid n} with τ ( n ) = σ 0 ( n ) = ∏ i = 1 n ( e i + 1 ) {\displaystyle \tau (n)=\sigma _{0}(n)=\prod _{i=1}^{n}(e_{i}+1)} for n = ∏ i = 1 n p i e i {\displaystyle n=\prod _{i=1}^{n}p_{i}^{e_{i}}} . The first few refactorable numbers are listed in (sequence A033950 in the OEIS) as

1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ... For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers.

Properties Cooper and Kennedy proved that refactorable numbers have natural density zero. Zelinsky proved that no three consecutive integers can all be refactorable. Colton proved that no refactorable number is perfect. The equation gcd ( n , x ) = τ ( n ) {\displaystyle \gcd(n,x)=\tau (n)} has solutions only if n {\displaystyle n} is a refactorable number, where gcd {\displaystyle \gcd } is the greatest common divisor function. Let T ( x ) {\displaystyle T(x)} be the number of refactorable numbers which are at most x {\displaystyle x} . The problem of determining an asymptotic for T ( x ) {\displaystyle T(x)} is open. Spiro has proven that T ( x ) = x log ⁡ x ( log ⁡ log ⁡ x ) 1 − o ( 1 ) {\displaystyle T(x)={\frac {x}{{\sqrt {\log x}}(\log \log x)^{1-o(1)}}}}

There are still unsolved problems regarding refactorable numbers. Colton asked if there are arbitrarily large n {\displaystyle n} such that both n {\displaystyle n} and n + 1 {\displaystyle n+1} are refactorable. Zelinsky wondered if there exists a refactorable number n 0 ≡ a mod m {\displaystyle n_{0}\equiv a\mod m} , does there necessarily exist n > n 0 {\displaystyle n>n_{0}} such that n {\displaystyle n} is refactorable and n ≡ a mod m {\displaystyle n\equiv a\mod m} .

History First defined by Curtis Cooper and Robert E. Kennedy where they showed that the tau numbers have natural density zero, they were later rediscovered by Simon Colton using a computer program he wrote ("HR") which invents and judges definitions from a variety of areas of mathematics such as number theory and graph theory. Colton called such numbers "refactorable". While computer programs had discovered proofs before, this discovery was one of the first times that a computer program had discovered a new or previously obscure idea. Colton proved many results about refactorable numbers, showing that there were infinitely many and proving a variety of congruence restrictions on their distribution. Colton was only later alerted that Kennedy and Cooper had previously investigated the topic.

Illustrations

Worked examples

Example 1 — a first encounter with Refactorable number

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

In research: Refactorable 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 Refactorable 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: Refactorable 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 Refactorable 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 Refactorable number in 20 minutes

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

Frequently asked questions

What is Refactorable number in simple terms?

Why does Refactorable 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 Refactorable 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 Refactorable number.