DivisorSigma[0, n] Sqrt[n^k] <= DivisorSigma[k, n] <= DivisorSigma[0, n] ((n^k + 1)/2) /; Element[k, Integers] && k >= 1

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 0 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <msup> <mi> n </mi> <mi> k </mi> </msup> </msqrt> </mrow> <mo> &#8804; </mo> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mi> k </mi> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mtext> </mtext> <mrow> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 0 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mi> n </mi> <mi> k </mi> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <leq /> <apply> <times /> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 0 </cn> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <ci> n </ci> <ci> k </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> DivisorSigma </ci> <ci> k </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 0 </cn> <ci> n </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> n </ci> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18