CarmichaelLambda[2^\[Alpha] Product[Subscript[p, k]^Subscript[\[Alpha], k], {k, 1, n}]] == LCM[CarmichaelLambda[2^\[Alpha]], CarmichaelLambda[Subscript[p, 1]^Subscript[\[Alpha], 1]], \[Ellipsis], CarmichaelLambda[Subscript[p, n]^Subscript[\[Alpha], n]]] /; Element[Subscript[p, k], Primes] && Element[\[Alpha], Integers] && \[Alpha] > 0 && Element[Subscript[\[Alpha], k], Integers] && Subscript[\[Alpha], k] > 0 && Subscript[p, k] > 2

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["CarmichaelLambda", "[", RowBox[List[SuperscriptBox["2", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], SubsuperscriptBox["p", "k", SubscriptBox["\[Alpha]", "k"]]]]]], "]"]], "\[Equal]", RowBox[List["LCM", "[", RowBox[List[RowBox[List["CarmichaelLambda", "[", SuperscriptBox["2", "\[Alpha]"], "]"]], ",", " ", RowBox[List["CarmichaelLambda", "[", SubsuperscriptBox["p", "1", SubscriptBox["\[Alpha]", "1"]], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["CarmichaelLambda", "[", SubsuperscriptBox["p", "n", SubscriptBox["\[Alpha]", "n"]], "]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["p", "k"], "\[Element]", "Primes"]], "\[And]", RowBox[List["\[Alpha]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Alpha]", ">", "0"]], "\[And]", RowBox[List[SubscriptBox["\[Alpha]", "k"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["\[Alpha]", "k"], ">", "0"]], "\[And]", RowBox[List[SubscriptBox["p", "k"], ">", "2"]]]]]]]]

<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> &#955; </mi> <annotation-xml encoding='MathML-Content'> <ci> CarmichaelLambda </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msubsup> <mi> p </mi> <mi> k </mi> <msub> <mi> &#945; </mi> <mi> k </mi> </msub> </msubsup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#955; </mi> <annotation-xml encoding='MathML-Content'> <ci> CarmichaelLambda </ci> </annotation-xml> </semantics> <mo> ( </mo> <msup> <mn> 2 </mn> <mi> &#945; </mi> </msup> <mo> ) </mo> </mrow> <mo> , </mo> <mtext> </mtext> <mrow> <semantics> <mi> &#955; </mi> <annotation-xml encoding='MathML-Content'> <ci> CarmichaelLambda </ci> </annotation-xml> </semantics> <mo> ( </mo> <msubsup> <mi> p </mi> <mn> 1 </mn> <msub> <mi> &#945; </mi> <mn> 1 </mn> </msub> </msubsup> <mo> ) </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <semantics> <mi> &#955; </mi> <annotation-xml encoding='MathML-Content'> <ci> CarmichaelLambda </ci> </annotation-xml> </semantics> <mo> ( </mo> <msubsup> <mi> p </mi> <mi> n </mi> <msub> <mi> &#945; </mi> <mi> n </mi> </msub> </msubsup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[Primes]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#945; </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> &#945; </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> &gt; </mo> <mn> 2 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> CarmichaelLambda </ci> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> &#945; </ci> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <lcm /> <apply> <ci> CarmichaelLambda </ci> <apply> <power /> <cn type='integer'> 2 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <ci> CarmichaelLambda </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> <apply> <ci> CarmichaelLambda </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> n </ci> </apply> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <primes /> </apply> <apply> <in /> <ci> &#945; </ci> <integers /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> k </ci> </apply> <integers /> </apply> <apply> <gt /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["CarmichaelLambda", "[", RowBox[List[SuperscriptBox["2", "\[Alpha]_"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n_"], SubsuperscriptBox["p_", "k", SubscriptBox["\[Alpha]_", "k"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["LCM", "[", RowBox[List[RowBox[List["CarmichaelLambda", "[", SuperscriptBox["2", "\[Alpha]"], "]"]], ",", RowBox[List["CarmichaelLambda", "[", SubsuperscriptBox["pp", "1", SubscriptBox["\[Alpha]\[Alpha]", "1"]], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["CarmichaelLambda", "[", SubsuperscriptBox["pp", "n", SubscriptBox["\[Alpha]\[Alpha]", "n"]], "]"]]]], "]"]], "/;", RowBox[List[RowBox[List[SubscriptBox["p", "k"], "\[Element]", "Primes"]], "&&", RowBox[List["\[Alpha]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Alpha]", ">", "0"]], "&&", RowBox[List[SubscriptBox["\[Alpha]", "k"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["\[Alpha]", "k"], ">", "0"]], "&&", RowBox[List[SubscriptBox["p", "k"], ">", "2"]]]]]]]]]]

2001-10-29