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Prime




Mathematica Notation

Traditional Notation









Number Theory Functions > Prime[n] > Series representations > Other series representations





http://functions.wolfram.com/13.03.06.0004.01









  


  








Input Form





Prime[n] \[Proportional] n (Log[n] + Log[Log[n]] - 1 + (Log[Log[n]] - 2)/Log[n] - (Log[Log[n]]^2 - 6 Log[Log[n]] + 11)/ (2 Log[n]^2) + \[Ellipsis]) /; n > 2








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Prime", "[", "n", "]"]], "\[Proportional]", RowBox[List["n", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "n", "]"]], "+", RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]], "-", "1", "+", FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]], "-", "2"]], RowBox[List["Log", "[", "n", "]"]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]], "2"], "-", RowBox[List["6", " ", RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]]]], "+", "11"]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["Log", "[", "n", "]"]], "2"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["n", ">", "2"]]]]]]








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> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <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> <mo> - </mo> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <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> <mo> - </mo> <mn> 2 </mn> </mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </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> <mn> 11 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 2 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> prime </ci> <ci> n </ci> </apply> <apply> <times /> <ci> n </ci> <apply> <plus /> <apply> <ln /> <ci> n </ci> </apply> <apply> <ln /> <apply> <ln /> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <ln /> <apply> <ln /> <ci> n </ci> </apply> </apply> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <apply> <ln /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <ln /> <apply> <ln /> <ci> n </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ln /> <apply> <ln /> <ci> n </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 11 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ln /> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Prime", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["n", " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "n", "]"]], "+", RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]], "-", "1", "+", FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]], "-", "2"]], RowBox[List["Log", "[", "n", "]"]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]], "2"], "-", RowBox[List["6", " ", RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]]]], "+", "11"]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["Log", "[", "n", "]"]], "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List["n", ">", "2"]]]]]]]]


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





2001-10-29