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In [[mathematics]], a '''sum-free sequence''' is an increasing [[positive integer]] [[sequence]]
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:<math>\{n_k\}_{k\in\mathbb N}</math>
 
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.
 
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.
 
==Example==
The [[power of two|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 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.
 
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>
 
==Density==
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.
 
==Notes==
{{reflist}}
 
==References==
*{{citation
| last = Abbott | first = H. L.
| issue = 1
| journal = Acta Arithmetica
| mr = 893466
| pages = 93–96
| title = On sum-free sequences
| volume = 48
| year = 1987}}.
*{{citation
| last1 = Deshouillers | first1 = Jean-Marc |authorlink1 = Jean-Marc Deshouillers
| last2 = Erdős | first2 = Pál | authorlink2 = Paul Erdős
| last3 = Melfi | first3 = Giuseppe | authorlink3 = Giuseppe Melfi
| pages = 49-54
| title = On a question about sum-free sequences
| journal = Discrete Mathematics
| mr = 1692278
| volume = 200
| year = 1999}}.
*{{citation
| last = Erdős | first = Pál | authorlink = Paul Erdős
| journal = Matematikai Lapok
| language = Hungarian
| mr = 0144871
| pages = 28–38
| title = Számelméleti megjegyzések, III. Néhány additív számelméleti problémáról | trans_chapter = Some remarks on number theory, III
| url = http://www.renyi.hu/~p_erdos/1962-22.pdf
| volume = 13
| year = 1962}}.
*{{citation
| last1 = Levine | first1 = Eugene
| last2 = O'Sullivan | first2 = Joseph
| issue = 1
| journal = Acta Arithmetica
| mr = 0466016
| pages = 9–24
| title = An upper estimate for the reciprocal sum of a sum-free sequence
| volume = 34
| year = 1977}}.
*{{citation
| last1 = Luczak | first1 = Tomasz
| last2 = Schoen | first2 = Tomasz
| journal = Acta Arithmetica
| mr =  1793162
| pages = 225–229
| volume = 95
| year = 2000}}.
*{{citation
| last = Yang | first = Shi Chun
| issue = 4
| journal = Journal of Mathematical Research and Exposition
| mr = 2549677
| pages = 753–755
| title = Note on the reciprocal sum of a sum-free sequence
| volume = 29
| year = 2009}}.
 
{{DEFAULTSORT:Sum-Free Sequence}}
[[Category:Additive combinatorics]]
[[Category:Integer sequences]]

Latest revision as of 02:52, 8 July 2014

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.