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GCD




Mathematica Notation

Traditional Notation









Integer Functions > GCD[n1,n2,...,nm] > Inequalities





http://functions.wolfram.com/04.08.29.0002.01









  


  








Input Form





Product[GCD[Subscript[n, Subscript[j, 1]], Subscript[n, Subscript[j, 2]], …, Subscript[n, Subscript[j, m]]], {Subscript[j, m], Subscript[j, m - 1] + 1, k}, …, {Subscript[j, 2], Subscript[j, 1] + 1, k}, {Subscript[j, 1], 1, k}] Product[LCM[Subscript[n, Subscript[j, 1]], Subscript[n, Subscript[j, 2]], …, Subscript[n, Subscript[j, m]]]^(m - 1), {Subscript[j, m], Subscript[j, m - 1] + 1, k}, …, {Subscript[j, 2], Subscript[j, 1] + 1, k} {Subscript[j, 1], 1, k}] >= Product[Subscript[n, j], {j, 1, k}]^ ((k - 1)!/((k - m)!*(m - 1)!)) >= Product[GCD[Subscript[n, Subscript[j, 1]], Subscript[n, Subscript[j, 2]], …, Subscript[n, Subscript[j, m]]]^(m - 1), {Subscript[j, m], Subscript[j, m - 1] + 1, k}, …, {Subscript[j, 2], Subscript[j, 1] + 1, k} {Subscript[j, 1], 1, k}]* Product[LCM[Subscript[n, Subscript[j, 1]], Subscript[n, Subscript[j, 2]], …, Subscript[n, Subscript[j, m]]], {Subscript[j, m], Subscript[j, m - 1] + 1, k}, …, {Subscript[j, 2], Subscript[j, 1] + 1, k} {Subscript[j, 1], 1, k}] /; Subscript[n, 1] ∈  && Subscript[n, 1] > 0 && Subscript[n, 2] ∈  && Subscript[n, 2] > 0 && … && Subscript[n, k] ∈  && Subscript[n, k] > 0 && k ∈  && k > 0 && m ∈  && 1 <= m <= k








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "1"], "=", "1"]], "k"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "2"], "=", RowBox[List[SubscriptBox["j", "1"], "+", "1"]]]], "k"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "m"], "=", RowBox[List[SubscriptBox["j", RowBox[List["m", "-", "1"]]], "+", "1"]]]], "k"], RowBox[List["GCD", "[", RowBox[List[SubscriptBox["n", SubscriptBox["j", "1"]], ",", SubscriptBox["n", SubscriptBox["j", "2"]], ",", "\[Ellipsis]", ",", SubscriptBox["n", SubscriptBox["j", "m"]]]], "]"]]]]]]]]]], " ", ")"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "1"], "=", "1"]], "k"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "2"], "=", RowBox[List[SubscriptBox["j", "1"], "+", "1"]]]], "k"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "m"], "=", RowBox[List[SubscriptBox["j", RowBox[List["m", "-", "1"]]], "+", "1"]]]], "k"], SuperscriptBox[RowBox[List["LCM", "[", RowBox[List[SubscriptBox["n", SubscriptBox["j", "1"]], ",", SubscriptBox["n", SubscriptBox["j", "2"]], ",", "\[Ellipsis]", ",", SubscriptBox["n", SubscriptBox["j", "m"]]]], "]"]], RowBox[List["m", "-", "1"]]]]]]]]]]], " ", ")"]]]], "\[GreaterEqual]", SuperscriptBox[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "k"], SubscriptBox["n", "j"]]], ")"]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]], "!"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]]]]], "\[GreaterEqual]", RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "1"], "=", "1"]], "k"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "2"], "=", RowBox[List[SubscriptBox["j", "1"], "+", "1"]]]], "k"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "m"], "=", RowBox[List[SubscriptBox["j", RowBox[List["m", "-", "1"]]], "+", "1"]]]], "k"], SuperscriptBox[RowBox[List["GCD", "[", RowBox[List[SubscriptBox["n", SubscriptBox["j", "1"]], ",", SubscriptBox["n", SubscriptBox["j", "2"]], ",", "\[Ellipsis]", ",", SubscriptBox["n", SubscriptBox["j", "m"]]]], "]"]], RowBox[List["m", "-", "1"]]]]]]]]]]], " ", ")"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "1"], "=", "1"]], "k"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "2"], "=", RowBox[List[SubscriptBox["j", "1"], "+", "1"]]]], "k"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List[SubscriptBox["j", "m"], "=", RowBox[List[SubscriptBox["j", RowBox[List["m", "-", "1"]]], "+", "1"]]]], "k"], RowBox[List["LCM", "[", RowBox[List[SubscriptBox["n", SubscriptBox["j", "1"]], ",", SubscriptBox["n", SubscriptBox["j", "2"]], ",", "\[Ellipsis]", ",", SubscriptBox["n", SubscriptBox["j", "m"]]]], "]"]]]]]]]]]], " ", ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["n", "1"], "\[Element]", "\[DoubleStruckCapitalZ]"]], "\[And]", RowBox[List[SubscriptBox["n", "1"], ">", "0"]], "\[And]", RowBox[List[SubscriptBox["n", "2"], "\[Element]", "\[DoubleStruckCapitalZ]"]], "\[And]", RowBox[List[SubscriptBox["n", "2"], ">", "0"]], "\[And]", "\[Ellipsis]", "\[And]", RowBox[List[SubscriptBox["n", "k"], "\[Element]", "\[DoubleStruckCapitalZ]"]], "\[And]", RowBox[List[SubscriptBox["n", "k"], ">", "0"]], "\[And]", RowBox[List["k", "\[Element]", "\[DoubleStruckCapitalZ]"]], "\[And]", RowBox[List["k", ">", "0"]], "\[And]", RowBox[List["m", "\[Element]", "\[DoubleStruckCapitalZ]"]], "\[And]", RowBox[List["1", "\[LessEqual]", "m", "\[LessEqual]", "k"]]]]]]]]








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> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> k </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mi> m </mi> </msub> <mo> = </mo> <mrow> <msub> <mi> j </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> k </mi> </munderover> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </msub> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> j </mi> <mi> m </mi> </msub> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> k </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mi> m </mi> </msub> <mo> = </mo> <mrow> <msub> <mi> j </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> k </mi> </munderover> <msup> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </msub> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> j </mi> <mi> m </mi> </msub> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8805; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <msub> <mi> n </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </msup> <mo> &#8805; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> k </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mi> m </mi> </msub> <mo> = </mo> <mrow> <msub> <mi> j </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> k </mi> </munderover> <msup> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </msub> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> j </mi> <mi> m </mi> </msub> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> k </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <msub> <mi> j </mi> <mi> m </mi> </msub> <mo> = </mo> <mrow> <msub> <mi> j </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> k </mi> </munderover> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </msub> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </mi> <msub> <mi> j </mi> <mi> m </mi> </msub> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mo> &#8230; </mo> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> m </mi> <mo> &#8804; </mo> <mi> k </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <geq /> <apply> <times /> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <ci> m </ci> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <gcd /> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <ci> m </ci> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <power /> <apply> <lcm /> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> </apply> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <ci> m </ci> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <power /> <apply> <gcd /> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <product /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <ci> m </ci> </apply> </bvar> <lowlimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <lcm /> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <ci> &#8230; </ci> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> k </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> m </ci> <ci> k </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2002-12-18