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| In [[mathematics]], a '''sum-free sequence''' is an increasing [[positive integer]] [[sequence]]
| | 35 yr old Leather Products Maker Merle from Roberval, has interests for example skateboarding, property developers in singapore and television watching. Will soon carry on [http://www.pediahub.com/early-days/govt-condominium/ buying a property in singapore] contiki voyage that will cover going to the Cidade Velha. |
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| :<math>\{n_k\}_{k\in\mathbb N}</math>
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| such that for each <math>k</math>, <math>n_k</math> cannot be represented as a sum of any subset of the preceding elements of the same sequence.
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| The definition of sum-free sequence is different of that of [[sum-free set]], because in a sum-free set only the sums of two elements must be avoided, while a sum-free sequence must avoid sums of larger sets of elements as well.
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| ==Example==
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| The [[power of two|powers of two]],
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| :1, 2, 4, 8, 16, ...
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| 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.
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| ==Sums of reciprocals==
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| A set of integers is said to be [[Small set (combinatorics)|small]] if the sum of its [[Multiplicative inverse|reciprocal]]s converges to a finite value. For instance, by the [[prime number theorem]], the [[prime number]]s are not small. {{harvs|first=Paul|last=Erdős|authorlink=Paul Erdős|year=1962|txt}} 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.
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| If <math>R</math> denotes the maximum sum of reciprocals of a sum-free sequence, then through subsequent research it is known that <math>2.065 < R < 3.97</math>.<ref>{{harvtxt|Levine|O'Sullivan|1977}}; {{harvtxt|Abbott|1987}}; {{harvtxt|Yang|2009}}.</ref>
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| ==Density==
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| It follows from the fact that sum-free sequences are small that they have zero [[Schnirelmann density]]; that is, if <math>A(x)</math> is defined to be the number of sequence elements that are less than or equal to <math>x</math>, then <math>A(x)=o(x)</math>. {{harvtxt|Erdős|1962}} showed that for every sum-free sequence there exists an unbounded sequence of numbers <math>x_i</math> for which <math>A(x_i)=O(x^{\varphi-1})</math> where <math>\varphi</math> is the [[golden ratio]], and he exhibited a sum-free sequence for which, for all values of <math>x</math>, <math>A(x)=\Omega(x^{2/7})</math>, subsequently improved to <math>A(x)=\Omega(x^{1/3})</math> by Deshouillers, Erdős and Melfi in 1999 and to <math>A(x)=\Omega(x^{1/2-\varepsilon})</math> by Luczak and Schoen in 2000, who also proved that the exponent ''1/2'' cannot be furhermore improved.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation
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| | last = Abbott | first = H. L.
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| | issue = 1
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| | journal = Acta Arithmetica
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| | mr = 893466
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| | pages = 93–96
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| | title = On sum-free sequences
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| | volume = 48
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| | year = 1987}}.
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| *{{citation
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| | last1 = Deshouillers | first1 = Jean-Marc |authorlink1 = Jean-Marc Deshouillers
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| | last2 = Erdős | first2 = Pál | authorlink2 = Paul Erdős
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| | last3 = Melfi | first3 = Giuseppe | authorlink3 = Giuseppe Melfi
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| | pages = 49-54
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| | title = On a question about sum-free sequences
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| | journal = Discrete Mathematics
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| | mr = 1692278
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| | volume = 200
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| | year = 1999}}.
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| *{{citation
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| | last = Erdős | first = Pál | authorlink = Paul Erdős
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| | journal = Matematikai Lapok
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| | language = Hungarian
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| | mr = 0144871
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| | pages = 28–38
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| | title = Számelméleti megjegyzések, III. Néhány additív számelméleti problémáról | trans_chapter = Some remarks on number theory, III
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| | url = http://www.renyi.hu/~p_erdos/1962-22.pdf
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| | volume = 13
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| | year = 1962}}.
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| *{{citation
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| | last1 = Levine | first1 = Eugene
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| | last2 = O'Sullivan | first2 = Joseph
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| | issue = 1
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| | journal = Acta Arithmetica
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| | mr = 0466016
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| | pages = 9–24
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| | title = An upper estimate for the reciprocal sum of a sum-free sequence
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| | volume = 34
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| | year = 1977}}.
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| *{{citation
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| | last1 = Luczak | first1 = Tomasz
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| | last2 = Schoen | first2 = Tomasz
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| | journal = Acta Arithmetica
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| | mr = 1793162
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| | pages = 225–229
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| | volume = 95
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| | year = 2000}}.
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| *{{citation
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| | last = Yang | first = Shi Chun
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| | issue = 4
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| | journal = Journal of Mathematical Research and Exposition
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| | mr = 2549677
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| | pages = 753–755
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| | title = Note on the reciprocal sum of a sum-free sequence
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| | volume = 29
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| | year = 2009}}.
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| {{DEFAULTSORT:Sum-Free Sequence}}
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| [[Category:Additive combinatorics]]
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| [[Category:Integer sequences]]
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35 yr old Leather Products Maker Merle from Roberval, has interests for example skateboarding, property developers in singapore and television watching. Will soon carry on buying a property in singapore contiki voyage that will cover going to the Cidade Velha.