System of bilinear equations: Difference between revisions

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In [[number theory]], an '''arithmetic number''' is an [[integer]] for which the [[arithmetic mean]] of its [[positive number|positive]] [[divisor]]s,  is an integer.  The first numbers in the [[sequence]] are 1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20 {{OEIS|id=A003601}}.  It is known that the [[natural density]] of such numbers is 1:<ref name=G76>Guy (2004) p.76</ref> indeed, the proportion of numbers less than ''X'' which are not arithmetic is [[Asymptotic analysis|asymptotically]]<ref name=BEPS>{{cite book | zbl=0478.10027 | last1=Bateman | first1=Paul T. | author1-link=Paul T. Bateman | last2=Erdős | first2=Paul | author2-link=Paul Erdős | last3=Pomerance | first3=Carl | author3-link=Carl Pomerance | last4=Straus | first4=E.G. | author4-link=Ernst G. Straus | chapter=The arithmetic mean of the divisors of an integer| title=Analytic number theory, Proc. Conf., Temple Univ., 1980 | editor-first=M.I. | editor-last=Knopp | editor-link=Marvin Knopp | series=Lecture Notes in Mathematics | volume=899 | pages=197–220 | year=1981 | publisher=[[Springer-Verlag]] | url = http://www.math.dartmouth.edu/~carlp/PDF/31.pdf}}</ref>
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:<math> \exp\left( { -c \sqrt{\log\log X} } \right) </math>
 
where ''c'' = 2 √ log 2 + o(1).
 
A number ''N'' is arithmetic if the [[number of divisors]] ''d''(''N'') divides the [[Sum-of-divisors function|sum of divisors]] σ(''N''). It is known that the [[Natural density|density]] of integers ''N'' for which ''d''(''N'')<sup>2</sup> divides  σ(''N'') is 1/2.<ref name=G76/><ref name=BEPS/>
 
==Notes==
{{reflist}}
 
== References ==
* {{cite book |last=Guy |first=Richard K. |authorlink=Richard K. Guy |title=Unsolved problems in number theory |publisher=[[Springer-Verlag]] |edition=3rd |year=2004 |isbn=978-0-387-20860-2 |zbl=1058.11001 |at=B2}}
 
{{Classes of natural numbers}}
{{Divisor classes}}
 
[[Category:Divisor function]]
[[Category:Integer sequences]]
[[Category:Unsolved problems in mathematics]]
 
 
{{numtheory-stub}}

Latest revision as of 12:54, 7 January 2015

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