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In mathematics, a [[integer sequence|sequence of positive integers]] ''a''''n'' is called an '''irrationality sequence''' if it has the property that, for every sequence ''x''''n'' of positive integers, the sum of the series

:<math> \sum_{n=1}^\infty \frac{1}{a_n x_n} </math>

exists and is an [[irrational number]].<ref name="guy">{{citation |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=0-387-20860-7 | zbl=1058.11001 | contribution=E24 Irrationality sequences|page=346|url=http://books.google.com/books?id=1AP2CEGxTkgC&pg=PA346 }}.</ref><ref>{{citation
| last1 = Erdős | first1 = P. | author1-link = Paul Erdős
| last2 = Graham | first2 = R. L. | author2-link = Ronald Graham
| location = Geneva
| mr = 592420
| page = 128
| publisher = Université de Genève L'Enseignement Mathématique
| series = Monographies de L'Enseignement Mathématique
| title = Old and new problems and results in combinatorial number theory
| volume = 28
| year = 1980}}.</ref> The problem of characterizing irrationality sequences was posed by [[Paul Erdős]] and [[Ernst G. Straus]], who originally called the property of being an irrationality sequence "Property P".<ref>{{citation
| last = Erdős | first = P. | authorlink = Paul Erdős
| journal = Journal of Mathematical Sciences
| mr = 539489
| pages = 1–7 (1976)
| title = Some problems and results on the irrationality of the sum of infinite series
| url = http://www.renyi.hu/~p_erdos/1976-44.pdf
| volume = 10
| year = 1975}}.</ref>

==Examples==
The [[Power of two#Powers of two whose exponents are powers of two|powers of two whose exponents are powers of two]], <math>2^{2^n}</math>, form an irrationality sequence. However, although [[Sylvester's sequence]]
:2, 3, 7, 43, 1807, 3263443, ...
(in which each term is one more than the product of all previous terms) also grows [[Double exponential function|doubly exponentially]], it does not form an irrationality sequence. For, letting <math>x_n=1</math> gives
:<math>\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots=1,</math>
a series converging to a rational number. Likewise, the [[factorial]]s <math>n!</math> do not form an irrationality sequence, because the sequence <math>x_n=n+2</math> leads to a series with a rational sum,
:<math>\sum_{n=0}^{\infty}\frac{1}{(n+2)n!}=\frac{1}{2}+\frac{1}{3}+\frac{1}{8}+\frac{1}{30}+\frac{1}{144}+\cdots=1.</math><ref name="guy"/>

==Growth rate==
Any sequence ''a''''n'' that grows at a rate such that
:<math>\limsup_n \frac{\log\log a_n}{n} > \log 2 </math>
is an irrationality sequence. This includes sequences that grow at a more than doubly exponential rate as well as some doubly exponential sequences that grow more quickly than the powers of powers of two.<ref name="guy"/>

Every irrationality sequence must grow quickly enough that
:<math>\lim_{n\to\infty} a_n^{1/n}=\infty.</math>
However, it is not known whether there exists such a sequence in which the [[greatest common divisor]] of each pair of terms is 1 (unlike the powers of powers of two) and for which
:<math>\lim_{n\to\infty} a_n^{1/2^n}<\infty.</math><ref>{{citation
| last = Erdős | first = P. | authorlink = Paul Erdős
| contribution = On the irrationality of certain series: problems and results
| location = Cambridge
| mr = 971997
| pages = 102–109
| publisher = Cambridge Univ. Press
| title = New advances in transcendence theory (Durham, 1986)
| url = http://www.renyi.hu/~p_erdos/1988-22.pdf
| year = 1988}}.</ref>

==Related properties==
Analogously to irrationality sequences,
{{harvtxt|Hančl|1996}} has defined a transcendental sequence to be an integer sequence ''a''''n'' such that, for every sequence ''x''''n'' of positive integers, the sum of the series
:<math> \sum_{n=1}^\infty \frac{1}{a_n x_n} </math>
exists and is an [[transcendental number]].<ref>{{citation
| last = Hančl | first = Jaroslav
| issue = 2-3
| journal = Mathematica Slovaca
| mr = 1427003
| pages = 177–179
| title = Transcendental sequences
| volume = 46
| year = 1996}}.</ref>

==References==
{{reflist}}

[[Category:Integer sequences]]
[[Category:Irrational numbers]]
[[Category:Number theory]]

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en>David Eppstein: link oxley