mathematics

Harmonic divisor number

Harmonic divisor 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 Harmonic divisor number rather than just read about it. In short: In mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS).

Key takeaways

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

Reference excerpt

In mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS). Harmonic divisor numbers were introduced by Øystein Ore, who showed that every perfect number is a harmonic divisor number and conjectured that there are no odd harmonic divisor numbers other than 1.

Examples The number 6 has four divisors: 1, 2, 3, and 6. Their harmonic mean is an integer:

4 1 1 + 1 2 + 1 3 + 1 6 = 2. {\displaystyle {\frac {4}{{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{6}}}}=2.}

Thus 6 is a harmonic divisor number. Similarly, the number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is

12 1 1 + 1 2 + 1 4 + 1 5 + 1 7 + 1 10 + 1 14 + 1 20 + 1 28 + 1 35 + 1 70 + 1 140 = 5. {\displaystyle {\frac {12}{{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{10}}+{\frac {1}{14}}+{\frac {1}{20}}+{\frac {1}{28}}+{\frac {1}{35}}+{\frac {1}{70}}+{\frac {1}{140}}}}=5.}

Since 5 is an integer, 140 is a harmonic divisor number.

Factorization of the harmonic mean The harmonic mean H(n) of the divisors of any number n can be expressed as the formula

H ( n ) = n σ 0 ( n ) σ 1 ( n ) {\displaystyle H(n)={\frac {n\sigma _{0}(n)}{\sigma _{1}(n)}}}

where σi (n) is the sum of ith powers of the divisors of n: σ0 is the number of divisors, and σ1 is the sum of divisors (Cohen 1997). All of the terms in this formula are multiplicative, so that the harmonic mean H(n) is also multiplicative. It follows that, for any positive integer n, the harmonic mean H(n) can be expressed as the product of the harmonic means of the prime powers in the factorization of n. For instance, we have

H ( 4 ) = 3 1 + 1 2 + 1 4 = 12 7 , {\displaystyle H(4)={\frac {3}{1+{\frac {1}{2}}+{\frac {1}{4}}}}={\frac {12}{7}},}

H ( 5 ) = 2 1 + 1 5 = 5 3 , {\displaystyle H(5)={\frac {2}{1+{\frac {1}{5}}}}={\frac {5}{3}},}

H ( 7 ) = 2 1 + 1 7 = 7 4 , {\displaystyle H(7)={\frac {2}{1+{\frac {1}{7}}}}={\frac {7}{4}},}

and

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Harmonic divisor number

Start with the simplest possible case. Write down what Harmonic divisor 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 Harmonic divisor 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 Harmonic divisor 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 Harmonic divisor number

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

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

Frequently asked questions

What is Harmonic divisor number in simple terms?

In mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS).

Why does Harmonic divisor 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 Harmonic divisor 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 Harmonic divisor number.