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Complete Sequence

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An increasing sequence of positive integers is a complete sequence if every term can be written as a sum of the first term, using the first term at most once [1].

A couple of variations on this definition:


  • Schissel [2] states that a complete sequence if every positive integer is a sum of “one or more distinct terms” in the sequence.
  • Erdos & Graham [3] describe a complete sequence as one where “every
    sufficiently largenatural number is a sum of distinct terms of the sequence. This particular definition is called a weakly complete sequence [4].
  • Linz & Jones [5] define a r-complete sequence, where every sufficiently large positive integer can be represented as the sum of r or more distinct terms from the sequence.

Complete Sequence Examples

A simple example of a complete sequence is {1, 2, 3, 4, …}.

Some complete sequences are more challenging to spot. For example, {1, 2, 3, 4, 8, 12, 16, 20, 24, 28, …} meets the definition because we can represent positive integers in modulo 4 (an arbitrary positive integer, a, can always be written as a = n * q + r).

The prime numbers are a complete sequence (if you add 1).

The Fibonacci sequence {1, 1, 2, 3, 5, 8, …} is an example of a complete sequence. Here [2],

  • fl, 1 (1) = 1, f1, 1(2) = 1, and
  • f1, 1(n) = f1, 1(n – 1) + f1, 1(n – 2) if n ≥ 3.

Removing a single number still leaves a complete sequence, although removing two numbers does not [6].

References

[1] Earl, R. (2017). Towards Higher Mathematics: A Companion. Cambridge University Press.
[2] Schissel , E. (!987). Characterizations of Three Types of Completeness. Retrieved April 7, 2021 from: https://www.fq.math.ca/Scanned/27-5/schissel.pdf
[3] Erdos, P. & Graham, R. (1980). Old and New Problems and Results in Combinatorial Number Theory: Monographie Numero 28 de L’Enseignement Math&matique .
Lf Enseignement Mathematique de 1’Universite de Geneve.
[4] Fox, A. & Knapp, M. (2013). A Note on Weakly Complete Sequences. Journal of Integer Sequences.
[5] Linz, W. & Jones, E. (2016). r-Completeness of Sequences of Positive Integers. Retrieved April 7, 2021 from: https://www.emis.de/journals/INTEGERS/papers/q59/q59.pdf
[6]Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985.

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Stephanie Glen. "Complete Sequence" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/complete-sequence/
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