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FactorInteger






Mathematica Notation

Traditional Notation









Number Theory Functions > FactorInteger[n] > Operations > Limit operation





http://functions.wolfram.com/13.01.25.0002.01









  


  










Input Form





Limit[(1/n) Sum[\[CapitalOmega][k], {k, 1, n}] - (Log[Log[n]] + c), n -> Infinity] == 0 /; (\[CapitalOmega][k] == Sum[Subscript[n, j], {j, 1, r}] /; k == Product[Prime[j]^Subscript[n, j], {j, 1, r}]) && c == EulerGamma + Sum[1/(Prime[k] (Prime[k] - 1)), {k, 1, Infinity}] + Sum[Log[1 - 1/Prime[k]] - 1/Prime[k], {k, 1, Infinity}]










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mi> n </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mi> &#937; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#937; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </munderover> <msub> <mi> n </mi> <mi> j </mi> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </munderover> <msup> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> j </mi> <mo> ) </mo> </mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> c </mi> <mo> &#10869; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> &#937; </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> c </ci> <apply> <ln /> <apply> <ln /> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> &#937; </ci> <ci> k </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> r </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> </apply> </apply> <apply> <eq /> <ci> k </ci> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> r </ci> </uplimit> <apply> <power /> <apply> <ci> prime </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <ci> c </ci> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <ci> prime </ci> <ci> k </ci> </apply> <apply> <plus /> <apply> <ci> prime </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> prime </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> prime </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <eulergamma /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2001-10-29





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