@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-P93ST75Z-8
  skos:prefLabel "théorie des nombres"@fr, "number theory"@en ;
  a skos:Concept ;
  skos:narrower psr:-TZSTCDZ4-K .

psr: a skos:ConceptScheme .
psr:-TZSTCDZ4-K
  skos:exactMatch <https://fr.wikipedia.org/wiki/Constante_de_Niven>, <https://en.wikipedia.org/wiki/Niven%27s_constant> ;
  skos:definition """En mathématiques, et plus précisément en théorie des nombres, la <b>constante de Niven</b>, portant le nom du mathématicien Ivan Niven, est la moyenne du plus grand exposant apparaissant dans la décomposition en produit de facteurs premiers d'un entier <span class="texhtml mvar" style="font-style:italic;">n</span>. Plus précisément, on définit <span class="texhtml mvar" style="font-style:italic;">H</span>(1) = 1 et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle H(n)=\\\\max _{p|n}v_{p}(n)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>H</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>n</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <munder>
<br/>          <mo movablelimits="true" form="prefix">max</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>p</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mo stretchy="false">|</mo>
<br/>            </mrow>
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </munder>
<br/>        <msub>
<br/>          <mi>v</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>p</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>n</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle H(n)=\\\\max _{p|n}v_{p}(n)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54a6985fa514551bfa0eb5f617f438fa7080908" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:18.47ex; height:4.509ex;" alt="{\\\\displaystyle H(n)=\\\\max _{p|n}v_{p}(n)}"></span> le plus grand exposant dans la décomposition en produit de facteurs premiers de <span class="texhtml mvar" style="font-style:italic;">n</span>&nbsp;&gt;&nbsp;1&nbsp;; la constante de Niven est définie par
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\lim _{n\\	o \\\\infty }{\\rac {1}{n}}\\\\sum _{j=1}^{n}H(j)=1+\\\\sum _{k=2}^{\\\\infty }\\\\left(1-{\\rac {1}{\\\\zeta (k)}}\\
ight)=1,705211\\\\dots }">
<br/>  <semantics>
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<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <munder>
<br/>          <mo movablelimits="true" form="prefix">lim</mo>
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<br/>            <mi>n</mi>
<br/>            <mo stretchy="false">→<!-- → --></mo>
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </munder>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mn>1</mn>
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<br/>            <mi>j</mi>
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<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
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<br/>        </munderover>
<br/>        <mi>H</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>j</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mn>1</mn>
<br/>        <mo>+</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
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<br/>            <mi>k</mi>
<br/>            <mo>=</mo>
<br/>            <mn>2</mn>
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<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
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<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <mn>1</mn>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mrow>
<br/>                  <mi>ζ<!-- ζ --></mi>
<br/>                  <mo stretchy="false">(</mo>
<br/>                  <mi>k</mi>
<br/>                  <mo stretchy="false">)</mo>
<br/>                </mrow>
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<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mn>1</mn>
<br/>        <mo>,</mo>
<br/>        <mn>705211</mn>
<br/>        <mo>…<!-- … --></mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lim _{n\\	o \\\\infty }{\\rac {1}{n}}\\\\sum _{j=1}^{n}H(j)=1+\\\\sum _{k=2}^{\\\\infty }\\\\left(1-{\\rac {1}{\\\\zeta (k)}}\\
ight)=1,705211\\\\dots }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cf3e0c4fca7e02e45247727ee1d54cbfe718413" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.338ex; width:54.45ex; height:7.176ex;" alt="{\\\\displaystyle \\\\lim _{n\\	o \\\\infty }{\\rac {1}{n}}\\\\sum _{j=1}^{n}H(j)=1+\\\\sum _{k=2}^{\\\\infty }\\\\left(1-{\\rac {1}{\\\\zeta (k)}}\\
ight)=1,705211\\\\dots }"></span></dd></dl>
<br/>où ζ(<i>k</i>) est la fonction zêta de Riemann au point <span class="texhtml mvar" style="font-style:italic;">k</span>. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Constante_de_Niven">https://fr.wikipedia.org/wiki/Constante_de_Niven</a>)"""@fr, """In number theory, <b>Niven's constant</b>, named after Ivan Niven, is the largest exponent appearing in the prime factorization of any natural number <i>n</i> "on average". More precisely, if we define <i>H</i>(1) = 1 and <i>H</i>(<i>n</i>) = the largest exponent appearing in the unique prime factorization of a natural number <i>n</i>&nbsp;&gt;&nbsp;1, then Niven's constant is given by
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\lim _{n\\	o \\\\infty }{\\rac {1}{n}}\\\\sum _{j=1}^{n}H(j)=1+\\\\sum _{k=2}^{\\\\infty }\\\\left(1-{\\rac {1}{\\\\zeta (k)}}\\
ight)=1.705211\\\\dots }">
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<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <munder>
<br/>          <mo movablelimits="true" form="prefix">lim</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo stretchy="false">→<!-- → --></mo>
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<br/>        <mrow class="MJX-TeXAtom-ORD">
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<br/>          <mo>∑<!-- ∑ --></mo>
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<br/>            <mi>j</mi>
<br/>            <mo>=</mo>
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<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
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<br/>        <mi>H</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>j</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mn>1</mn>
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<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>            <mo>=</mo>
<br/>            <mn>2</mn>
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<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
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<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <mn>1</mn>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mrow>
<br/>                  <mi>ζ<!-- ζ --></mi>
<br/>                  <mo stretchy="false">(</mo>
<br/>                  <mi>k</mi>
<br/>                  <mo stretchy="false">)</mo>
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<br/>              </mfrac>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mo>)</mo>
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<br/>        <mn>1.705211</mn>
<br/>        <mo>…<!-- … --></mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lim _{n\\	o \\\\infty }{\\rac {1}{n}}\\\\sum _{j=1}^{n}H(j)=1+\\\\sum _{k=2}^{\\\\infty }\\\\left(1-{\\rac {1}{\\\\zeta (k)}}\\
ight)=1.705211\\\\dots }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe5d8c2329f66b2814b8950768a4824de31e1d5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.338ex; width:54.063ex; height:7.176ex;" alt="{\\\\displaystyle \\\\lim _{n\\	o \\\\infty }{\\rac {1}{n}}\\\\sum _{j=1}^{n}H(j)=1+\\\\sum _{k=2}^{\\\\infty }\\\\left(1-{\\rac {1}{\\\\zeta (k)}}\\
ight)=1.705211\\\\dots }"></span></dd></dl>
<br/>where ζ is the Riemann zeta function.
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Niven%27s_constant">https://en.wikipedia.org/wiki/Niven%27s_constant</a>)"""@en ;
  skos:inScheme psr: ;
  skos:broader psr:-P93ST75Z-8, psr:-RBFVN7DN-2 ;
  dc:modified "2023-08-03"^^xsd:date ;
  dc:created "2023-08-03"^^xsd:date ;
  a skos:Concept ;
  skos:prefLabel "Niven's constant"@en, "constante de Niven"@fr .

psr:-RBFVN7DN-2
  skos:prefLabel "mathematical constant"@en, "constante mathématique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-TZSTCDZ4-K .