RamanujanTau[n Subscript[p, j]^m] == RamanujanTau[Subscript[p, j]] RamanujanTau[n Subscript[p, j]^(m - 1)] - Subscript[p, j]^11 RamanujanTau[n Subscript[p, j]^(m - 2)] /; Element[n, Integers] && n > 0 && Element[m, Integers] && m >= 2 && GCD[n, Subscript[p, j]] == 1 && Element[Subscript[p, j], Primes]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["RamanujanTau", "[", RowBox[List["n", " ", SubsuperscriptBox["p", "j", "m"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["RamanujanTau", "[", SubscriptBox["p", "j"], "]"]], RowBox[List["RamanujanTau", "[", RowBox[List["n", " ", SubsuperscriptBox["p", "j", RowBox[List["m", "-", "1"]]]]], "]"]]]], "-", RowBox[List[SubsuperscriptBox["p", "j", "11"], RowBox[List["RamanujanTau", "[", RowBox[List["n", " ", SubsuperscriptBox["p", "j", RowBox[List["m", "-", "2"]]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "2"]], "\[And]", RowBox[List[RowBox[List["GCD", "[", RowBox[List["n", ",", SubscriptBox["p", "j"]]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", "j"], "\[Element]", "Primes"]]]]]]]]

<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> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msubsup> <mi> p </mi> <mi> j </mi> <mi> m </mi> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mrow> <semantics> <mi> &#964; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Tau]&quot;, RamanujanTau] </annotation> </semantics> <mo> ( </mo> <msub> <mi> p </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#964; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Tau]&quot;, RamanujanTau] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msubsup> <mi> p </mi> <mi> j </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msubsup> <mi> p </mi> <mi> j </mi> <mn> 11 </mn> </msubsup> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#964; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Tau]&quot;, RamanujanTau] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msubsup> <mi> p </mi> <mi> j </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8805; </mo> <mn> 2 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <msub> <mi> p </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[List[], Primes]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> RamanujanTau </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> RamanujanTau </ci> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> </apply> <apply> <ci> RamanujanTau </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <cn type='integer'> 11 </cn> </apply> <apply> <ci> RamanujanTau </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <geq /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <eq /> <apply> <gcd /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <primes /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["RamanujanTau", "[", RowBox[List["n_", " ", SubsuperscriptBox["p_", "j", "m_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["RamanujanTau", "[", SubscriptBox["p", "j"], "]"]], " ", RowBox[List["RamanujanTau", "[", RowBox[List["n", " ", SubsuperscriptBox["p", "j", RowBox[List["m", "-", "1"]]]]], "]"]]]], "-", RowBox[List[SubsuperscriptBox["p", "j", "11"], " ", RowBox[List["RamanujanTau", "[", RowBox[List["n", " ", SubsuperscriptBox["p", "j", RowBox[List["m", "-", "2"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "2"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["n", ",", SubscriptBox["p", "j"]]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", "j"], "\[Element]", "Primes"]]]]]]]]]]

2007-05-02