DivisorSigma[1, n] <= HarmonicNumber[n] + Exp[HarmonicNumber[n]] Log[HarmonicNumber[n]] /; Element[n, Integers] && n >= 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> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 1 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mrow> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> </msup> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <leq /> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 1 </cn> <ci> n </ci> </apply> <apply> <plus /> <apply> <ci> HarmonicNumber </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <ci> HarmonicNumber </ci> <ci> n </ci> </apply> </apply> <apply> <ln /> <apply> <ci> HarmonicNumber </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["1", ",", "n"]], "]"]], "\[LessEqual]", RowBox[List[RowBox[List["HarmonicNumber", "[", "n", "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["HarmonicNumber", "[", "n", "]"]]], " ", RowBox[List["Log", "[", RowBox[List["HarmonicNumber", "[", "n", "]"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]]]]]]]]

2002-12-18