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Prime






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/13.03.06.0003.01









  


  










Input Form





Prime[n] == Sum[g[1 - g[Sum[r[g[j - 1]!^2, j], {j, 0, k}] - n]], {k, 0, n^2}] /; g[m] == m UnitStep[m] && r[a, b] == a KroneckerDelta[b, 0] + (1 - KroneckerDelta[b, 0]) Mod[a, b]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Prime", "[", "n", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], SuperscriptBox["n", "2"]], RowBox[List["g", "[", RowBox[List["1", "-", RowBox[List["g", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List["r", "[", RowBox[List[SuperscriptBox[RowBox[List[RowBox[List["g", "[", RowBox[List["j", "-", "1"]], "]"]], "!"]], "2"], ",", "j"]], "]"]]]], "-", "n"]], "]"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["g", "[", "m", "]"]], "\[Equal]", RowBox[List["m", " ", RowBox[List["UnitStep", "[", "m", "]"]]]]]], "\[And]", RowBox[List[RowBox[List["r", "[", RowBox[List["a", ",", "b"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["a", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["b", ",", "0"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["KroneckerDelta", "[", RowBox[List["b", ",", "0"]], "]"]]]], ")"]], " ", RowBox[List["Mod", "[", RowBox[List["a", ",", "b"]], "]"]]]]]]]]]]]]]]










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> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </munderover> <mrow> <mi> g </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <mi> r </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mn> 2 </mn> </msup> <mo> , </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> r </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> b </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> b </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <semantics> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> FE`Conversion`Private`a </ci> <ci> FE`Conversion`Private`b </ci> </apply> </annotation-xml> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> prime </ci> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </uplimit> <apply> <times /> <ci> g </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> g </ci> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <ci> r </ci> <apply> <power /> <apply> <factorial /> <apply> <ci> g </ci> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <ci> j </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> g </ci> <ci> m </ci> </apply> <apply> <times /> <ci> m </ci> <apply> <ci> UnitStep </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> r </ci> <ci> a </ci> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <ci> KroneckerDelta </ci> <ci> b </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KroneckerDelta </ci> <ci> b </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <rem /> <ci> FE`Conversion`Private`a </ci> <ci> FE`Conversion`Private`b </ci> </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[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], SuperscriptBox["n", "2"]], RowBox[List["g", "[", RowBox[List["1", "-", RowBox[List["g", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List["r", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["g", "[", RowBox[List["j", "-", "1"]], "]"]], "!"]], ")"]], "2"], ",", "j"]], "]"]]]], "-", "n"]], "]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["g", "[", "m", "]"]], "\[Equal]", RowBox[List["m", " ", RowBox[List["UnitStep", "[", "m", "]"]]]]]], "&&", RowBox[List[RowBox[List["r", "[", RowBox[List["a", ",", "b"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["a", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["b", ",", "0"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["KroneckerDelta", "[", RowBox[List["b", ",", "0"]], "]"]]]], ")"]], " ", RowBox[List["Mod", "[", RowBox[List["a", ",", "b"]], "]"]]]]]]]]]]]]]]]]










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





2001-10-29





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