en>RJFJR: remove {{cleanup|date=July 2008}}

2014-07-08T00:52:24Z

remove {{cleanup|date=July 2008}}

← Older revision		Revision as of 02:52, 8 July 2014
Line 1:		Line 1:
	~~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.

	~~:<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]]~~

en>Doctorhook: Fixed union notation

2012-11-22T21:54:06Z

Fixed union notation

New page

In [[mathematics]], a '''sum-free sequence''' is an increasing [[positive integer]] [[sequence]]

:<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]]

← Older revision		Revision as of 02:52, 8 July 2014
Line 1:		Line 1:
	~~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.

	~~:<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]]~~

Hyperstructure - Revision history

en>RJFJR: remove {{cleanup|date=July 2008}}

en>Doctorhook: Fixed union notation