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Prime




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/13.03.06.0002.01









  


  








Input Form





Prime[n] == Sum[ m Floor[1/(1 + Abs[n - Floor[1/Sum[Floor[Floor[m/i]/(m/i)], {i, 1, m - 1}]] Sum[Floor[1/Sum[Floor[Floor[k/i]/(k/i)], {i, 1, k - 1}]], {k, 2, m}]])], {m, 2, 2^n}]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Prime", "[", "n", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "2"]], SuperscriptBox["2", "n"]], RowBox[List["m", " ", RowBox[List["Floor", "[", RowBox[List["1", "/", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Abs", "[", RowBox[List["n", "-", RowBox[List[RowBox[List["Floor", "[", FractionBox["1", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], RowBox[List["m", "-", "1"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Floor", "[", FractionBox["m", "i"], "]"]], FractionBox["m", "i"]], "]"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "m"], RowBox[List["Floor", "[", FractionBox["1", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Floor", "[", FractionBox["k", "i"], "]"]], FractionBox["k", "i"]], "]"]]]]], "]"]]]]]]]], "]"]]]], ")"]]]], "]"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <msup> <mn> 2 </mn> <mi> n </mi> </msup> </munderover> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mn> 1 </mn> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mn> 1 </mn> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mo> &#8970; </mo> <mfrac> <mi> m </mi> <mi> i </mi> </mfrac> <mo> &#8971; </mo> </mrow> <mfrac> <mi> m </mi> <mi> i </mi> </mfrac> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mo> &#8970; </mo> <mfrac> <mn> 1 </mn> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mo> &#8970; </mo> <mfrac> <mi> k </mi> <mi> i </mi> </mfrac> <mo> &#8971; </mo> </mrow> <mfrac> <mi> k </mi> <mi> i </mi> </mfrac> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mrow> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> prime </ci> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </uplimit> <apply> <times /> <ci> m </ci> <apply> <floor /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <abs /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <floor /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <floor /> <apply> <times /> <apply> <floor /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> i </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> i </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <floor /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <floor /> <apply> <times /> <apply> <floor /> <apply> <times /> <ci> k </ci> <apply> <power /> <ci> i </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <power /> <ci> i </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Prime", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "2"]], SuperscriptBox["2", "n"]], RowBox[List["m", " ", RowBox[List["Floor", "[", FractionBox["1", RowBox[List["1", "+", RowBox[List["Abs", "[", RowBox[List["n", "-", RowBox[List[RowBox[List["Floor", "[", FractionBox["1", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], RowBox[List["m", "-", "1"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Floor", "[", FractionBox["m", "i"], "]"]], FractionBox["m", "i"]], "]"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "m"], RowBox[List["Floor", "[", FractionBox["1", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Floor", "[", FractionBox["k", "i"], "]"]], FractionBox["k", "i"]], "]"]]]]], "]"]]]]]]]], "]"]]]]], "]"]]]]]]]]]]


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





2001-10-29