Fibonacci[n] == Limit[Max[Table[Log[Log[Subscript[d, n - 1][μ]]]/ Log[Log[μ]],{μ,1,m}]], m -> Infinity] /; n ∈ Integers && n >= 1 && Subscript[d, k][m] == Subscript[d, k - 1][DivisorSigma[0, m]] && Subscript[d, 0][m] == DivisorSigma[0, m]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Fibonacci", "[", "n", "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List["Max", "[", SubscriptBox[RowBox[List["{", " ", FractionBox[RowBox[List["Log", "[", RowBox[List["Log", "[", RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "1"]]], "[", "\[Mu]", "]"]], " ", "]"]], "]"]], RowBox[List["Log", "[", RowBox[List["Log", "[", "\[Mu]", "]"]], "]"]]], "}"]], RowBox[List["\[Mu]", ",", "1", ",", "m"]]], "]"]], ",", RowBox[List["m", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "1"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["d", "k"], "[", "m", "]"]], "\[Equal]", RowBox[List[SubscriptBox["d", RowBox[List["k", "-", "1"]]], "[", RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", "m"]], "]"]], "]"]]]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["d", "0"], "[", "m", "]"]], "\[Equal]", RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", "m"]], "]"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> F </mi> <annotation encoding='Mathematica'> TagBox[&quot;F&quot;, Fibonacci] </annotation> </semantics> <mi> n </mi> </msub> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> m </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mi> max </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mrow> <mo> { </mo> <mfrac> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> d </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> &#956; </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#956; </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> } </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mi> m </mi> </mrow> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msub> <mi> d </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 0 </mn> </msub> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> d </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 0 </mn> </msub> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Fibonacci </ci> <ci> n </ci> </apply> <apply> <limit /> <bvar> <ci> m </ci> </bvar> <condition> <apply> <tendsto /> <ci> m </ci> <infinity /> </apply> </condition> <apply> <max /> <apply> <ci> Subscript </ci> <list> <apply> <times /> <apply> <ln /> <apply> <ln /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#956; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ln /> <apply> <ln /> <ci> &#956; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </list> <ci> &#956; </ci> <cn type='integer'> 1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <ci> m </ci> </apply> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 0 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 0 </cn> </apply> <ci> m </ci> </apply> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 0 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Fibonacci", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["Max", "[", SubscriptBox[RowBox[List["{", FractionBox[RowBox[List["Log", "[", RowBox[List["Log", "[", RowBox[List[SubscriptBox["d", RowBox[List["n", "-", "1"]]], "[", "\[Mu]", "]"]], "]"]], "]"]], RowBox[List["Log", "[", RowBox[List["Log", "[", "\[Mu]", "]"]], "]"]]], "}"]], RowBox[List["\[Mu]", ",", "1", ",", "m"]]], "]"]], ",", RowBox[List["m", "\[Rule]", "\[Infinity]"]]]], "]"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["d", "k"], "[", "m", "]"]], "\[Equal]", RowBox[List[SubscriptBox["d", RowBox[List["k", "-", "1"]]], "[", RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", "m"]], "]"]], "]"]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["d", "0"], "[", "m", "]"]], "\[Equal]", RowBox[List["DivisorSigma", "[", RowBox[List["0", ",", "m"]], "]"]]]]]]]]]]]]

2002-12-18