science

Sum-free sequence

Sum-free sequence is a science 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 Sum-free sequence rather than just read about it. In short: In mathematics, a sum-free sequence is an increasing sequence of positive integers, a 1 , a 2 , a 3 , … , {\displaystyle a_{1},a_{2},a_{3},\ldots ,} such that no term a n {\displaystyle a_{n}} can be represented as a sum of any subset of the preceding elements of the sequence. This differs from a sum-free set, where only pairs of sums must be avoided, but where those sums may come from the whole set rather than just…

Key takeaways

Sum-free sequence belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sum-free sequence to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sum-free sequence from memory before moving on to harder problems.

Reference excerpt

In mathematics, a sum-free sequence is an increasing sequence of positive integers,

a 1 , a 2 , a 3 , … , {\displaystyle a_{1},a_{2},a_{3},\ldots ,}

such that no term a n {\displaystyle a_{n}} can be represented as a sum of any subset of the preceding elements of the sequence. This differs from a sum-free set, where only pairs of sums must be avoided, but where those sums may come from the whole set rather than just the preceding terms.

Example The powers of two,

1, 2, 4, 8, 16, ... form a sum-free sequence: each term in the sequence is one more than the sum of all preceding terms, and so cannot be represented as a sum of preceding terms.

Sums of reciprocals A set of integers is said to be small if the sum of its reciprocals converges to a finite value. For instance, by the prime number theorem, the prime numbers are not small. Paul Erdős (1962) proved that every sum-free sequence is small, and asked how large the sum of reciprocals could be. For instance, the sum of the reciprocals of the powers of two (a geometric series) is two. If R {\displaystyle R} denotes the supremum of all sums of reciprocals of a sum-free sequence, then through subsequent research it is known that 2.0654 < R < 2.8570 {\displaystyle 2.0654<R<2.8570} .

Density It follows from the fact that sum-free sequences are small that they have zero Schnirelmann density; that is, if A ( x ) {\displaystyle A(x)} is defined to be the number of sequence elements that are less than or equal to x {\displaystyle x} , then A ( x ) = o ( x ) {\displaystyle A(x)=o(x)} . Erdős (1962) showed that for every sum-free sequence there exists an unbounded sequence of numbers x i {\displaystyle x_{i}} for which A ( x i ) = O ( x φ − 1 ) {\displaystyle A(x_{i})=O(x^{\varphi -1})} where φ {\displaystyle \varphi } is the golden ratio, and he exhibited a sum-free sequence for which, for all values of x {\displaystyle x} , A ( x ) = Ω ( x 2 / 7 ) {\displaystyle A(x)=\Omega (x^{2/7})} , subsequently improved to A ( x ) = Ω ( x 1 / 3 ) {\displaystyle A(x)=\Omega (x^{1/3})} by Deshouillers, Erdős and Melfi in 1999 and to A ( x ) = Ω ( x 1 / 2 − ε ) {\displaystyle A(x)=\Omega (x^{1/2-\varepsilon })} by Luczak and Schoen in 2000, who also proved that the exponent 1/2 cannot be further improved.

Notes

References Abbott, H. L. (1987), "On sum-free sequences", Acta Arithmetica, 48 (1): 93–96, doi:10.4064/aa-48-1-93-96, MR 0893466. Chen, Yong Gao (2013), "On the reciprocal sum of a sum-free sequence", Science China Mathematics, 56 (5): 951–966, Bibcode:2013ScChA..56..951C, doi:10.1007/s11425-012-4540-6, S2CID 124005748. Deshouillers, Jean-Marc; Erdős, Pál; Melfi, Giuseppe (1999), "On a question about sum-free sequences", Discrete Mathematics, 200 (1–3): 49–54, doi:10.1016/s0012-365x(98)00322-7, MR 1692278. Erdős, Pál (1962), "Számelméleti megjegyzések, III. Néhány additív számelméleti problémáról" [Some remarks on number theory, III] (PDF), Matematikai Lapok (in Hungarian), 13: 28–38, MR 0144871. Levine, Eugene; O'Sullivan, Joseph (1977), "An upper estimate for the reciprocal sum of a sum-free sequence", Acta Arithmetica, 34 (1): 9–24, doi:10.4064/aa-34-1-9-24, MR 0466016. Luczak, Tomasz; Schoen, Tomasz (2000), "On the maximal density of sum-free sets", Acta Arithmetica, 95 (3): 225–229, doi:10.4064/aa-95-3-225-229, MR 1793162. Yang, Shi Chun (2009), "Note on the reciprocal sum of a sum-free sequence", Journal of Mathematical Research and Exposition, 29 (4): 753–755, MR 2549677. Yang, Shi Chun (2015), "An upper bound for Erdös reciprocal sum of the sum-free sequence", Scientia Sinica Mathematica, 45 (3): 213–232, doi:10.1360/N012014-00121.

Worked examples

Example 1 — a first encounter with Sum-free sequence

Start with the simplest possible case. Write down what Sum-free sequence claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Sum-free sequence 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 Sum-free sequence 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 Sum-free sequence

In research: Sum-free sequence appears in science 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 Sum-free sequence 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: Sum-free sequence is common in secondary-school and first-year university syllabi. It links to neighbouring topics Additive combinatorics, Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Sum-free sequence 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 Sum-free sequence in 20 minutes

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

Frequently asked questions

What is Sum-free sequence in simple terms?

In mathematics, a sum-free sequence is an increasing sequence of positive integers, a 1 , a 2 , a 3 , … , {\displaystyle a_{1},a_{2},a_{3},\ldots ,} such that no term a n {\displaystyle a_{n}} can be represented as a sum of any subset of the preceding elements of the sequence. This differs from a s…

Why does Sum-free sequence matter?

Because it connects several science 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 Sum-free sequence?

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 Sum-free sequence.