PrimePi[x] == Sum[1 + Floor[(2 - Sum[Floor[j/i] - Floor[(j - 1)/i], {i, 1, j}])/j], {j, 2, Floor[x]}]

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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> <semantics> <mi> &#960; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Pi]&quot;, PrimePi] </annotation> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mi> x </mi> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> j </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mi> j </mi> <mi> i </mi> </mfrac> <mo> &#8971; </mo> </mrow> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> i </mi> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mi> j </mi> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> PrimePi </ci> <ci> x </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <apply> <floor /> <ci> x </ci> </apply> </uplimit> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> j </ci> </uplimit> <apply> <plus /> <apply> <floor /> <apply> <times /> <ci> j </ci> <apply> <power /> <ci> i </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> i </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18