mathematics

Sparsely totient number

Sparsely totient 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 Sparsely totient number rather than just read about it. In short: In mathematics, specifically number theory, a sparsely totient number is a natural number, n, such that for all m > n, φ ( m ) > φ ( n ) {\displaystyle \varphi (m)>\varphi (n)} where φ {\displaystyle \varphi } is Euler's totient function. The first few sparsely totient numbers are: 2, 6, 12, 18, 30, 42, 60, 66, 90, 120, 126, 150, 210, 240, 270, 330, 420, 462, 510, 630, 660, 690, 840, 870, 1050, 1260, 1320, 1470, 168…

Key takeaways

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

Reference excerpt

In mathematics, specifically number theory, a sparsely totient number is a natural number, n, such that for all m > n,

φ ( m ) > φ ( n ) {\displaystyle \varphi (m)>\varphi (n)}

where φ {\displaystyle \varphi } is Euler's totient function. The first few sparsely totient numbers are: 2, 6, 12, 18, 30, 42, 60, 66, 90, 120, 126, 150, 210, 240, 270, 330, 420, 462, 510, 630, 660, 690, 840, 870, 1050, 1260, 1320, 1470, 1680, 1890, 2310, 2730, 2940, 3150, 3570, 3990, 4620, 4830, 5460, 5610, 5670, 6090, 6930, 7140, 7350, 8190, 9240, 9660, 9870, ... (sequence A036913 in the OEIS). The concept was introduced by David Masser and Peter Man-Kit Shiu in 1986. As they showed, every primorial is sparsely totient.

Properties If P(n) is the largest prime factor of n, then lim inf P ( n ) / log ⁡ n = 1 {\displaystyle \liminf P(n)/\log n=1} .

P ( n ) ≪ log δ ⁡ n {\displaystyle P(n)\ll \log ^{\delta }n} holds for an exponent δ = 37 / 20 {\displaystyle \delta =37/20} . It is conjectured that lim sup P ( n ) / log ⁡ n = 2 {\displaystyle \limsup P(n)/\log n=2} . They are always even because if x is odd, then 2x also has the same Totient function, trivially failing the condition that all numbers more than it has more value of Totient function than it.

References

Baker, Roger C.; Harman, Glyn (1996). "Sparsely totient numbers". Annales de la Faculté des Sciences de Toulouse: Mathématiques. 5 (2): 183–190. doi:10.5802/afst.826 (inactive 10 October 2025). ISSN 0240-2963. Zbl 0871.11060.{{cite journal}}: CS1 maint: DOI inactive as of October 2025 (link) Masser, D.W.; Shiu, P. (1986). "On sparsely totient numbers". Pacific Journal of Mathematics. 121 (2): 407–426. doi:10.2140/pjm.1986.121.407. ISSN 0030-8730. MR 0819198. S2CID 55350630. Zbl 0538.10006.

Worked examples

Example 1 — a first encounter with Sparsely totient number

Start with the simplest possible case. Write down what Sparsely totient 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 Sparsely totient 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 Sparsely totient 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 Sparsely totient number

In research: Sparsely totient 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 Sparsely totient 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: Sparsely totient 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 Sparsely totient 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 Sparsely totient number in 20 minutes

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

Frequently asked questions

What is Sparsely totient number in simple terms?

In mathematics, specifically number theory, a sparsely totient number is a natural number, n, such that for all m > n, φ ( m ) > φ ( n ) {\displaystyle \varphi (m)>\varphi (n)} where φ {\displaystyle \varphi } is Euler's totient function. The first few sparsely totient numbers are: 2, 6, 12, 18, 30…

Why does Sparsely totient 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 Sparsely totient 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 Sparsely totient number.