RamanujanTau[n] == (691/1800) DivisorSigma[3, n] + (691/900) DivisorSigma[7, n] - (91/600) DivisorSigma[11, n] + (2764/15) Sum[DivisorSigma[3, k] DivisorSigma[7, n - k], {k, 1, n - 1}] /; n > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RamanujanTau", "[", "n", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["691", "1800"], RowBox[List["DivisorSigma", "[", RowBox[List["3", ",", "n"]], "]"]]]], "+", RowBox[List[FractionBox["691", "900"], RowBox[List["DivisorSigma", "[", RowBox[List["7", ",", "n"]], "]"]]]], "-", RowBox[List[FractionBox["91", "600"], RowBox[List["DivisorSigma", "[", RowBox[List["11", ",", "n"]], "]"]]]], "+", RowBox[List[FractionBox["2764", "15"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["3", ",", "k"]], "]"]], RowBox[List["DivisorSigma", "[", RowBox[List["7", ",", RowBox[List["n", "-", "k"]]]], "]"]]]]]]]]]]]], "/;", RowBox[List["n", ">", "0"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mi> &#964; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Tau]&quot;, RamanujanTau] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mn> 691 </mn> <mn> 1800 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 691 </mn> <mn> 900 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 7 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 91 </mn> <mn> 600 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 11 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 2764 </mn> <mn> 15 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 3 </mn> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 7 </mn> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> RamanujanTau </ci> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 691 <sep /> 1800 </cn> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 691 <sep /> 900 </cn> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 7 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 91 <sep /> 600 </cn> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 11 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 2764 <sep /> 15 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 3 </cn> <ci> k </ci> </apply> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 7 </cn> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RamanujanTau", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["691", " ", RowBox[List["DivisorSigma", "[", RowBox[List["3", ",", "n"]], "]"]]]], "1800"], "+", RowBox[List[FractionBox["691", "900"], " ", RowBox[List["DivisorSigma", "[", RowBox[List["7", ",", "n"]], "]"]]]], "-", RowBox[List[FractionBox["91", "600"], " ", RowBox[List["DivisorSigma", "[", RowBox[List["11", ",", "n"]], "]"]]]], "+", RowBox[List[FractionBox["2764", "15"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["3", ",", "k"]], "]"]], " ", RowBox[List["DivisorSigma", "[", RowBox[List["7", ",", RowBox[List["n", "-", "k"]]]], "]"]]]]]]]]]], "/;", RowBox[List["n", ">", "0"]]]]]]]]

2007-05-02