@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: a skos:ConceptScheme .
psr:-CVDPQB0Q-M
  skos:prefLabel "natural numbers"@en, "entier naturel"@fr ;
  a skos:Concept ;
  skos:narrower psr:-V2HDFL9H-Z .

psr:-V2HDFL9H-Z
  skos:definition """In mathematics, a <b>superabundant number</b> is a certain kind of natural number. A natural number <span class="texhtml mvar" style="font-style:italic;">n</span> is called <i>superabundant</i> precisely when, for all <span class="texhtml"><i>m</i> &lt; <i>n</i></span>:  <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 {\\rac {\\\\sigma (m)}{m}}<{\\rac {\\\\sigma (n)}{n}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mi>σ<!-- σ --></mi>               <mo stretchy="false">(</mo>               <mi>m</mi>               <mo stretchy="false">)</mo>             </mrow>             <mi>m</mi>           </mfrac>         </mrow>         <mo>&lt;</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mi>σ<!-- σ --></mi>               <mo stretchy="false">(</mo>               <mi>n</mi>               <mo stretchy="false">)</mo>             </mrow>             <mi>n</mi>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {\\\\sigma (m)}{m}}&lt;{\\rac {\\\\sigma (n)}{n}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60bb796902062647cd1d0e2e48615d955816e210" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.484ex; height:5.676ex;" alt="{\\rac  {\\\\sigma (m)}{m}}<{\\rac  {\\\\sigma (n)}{n}}"></span></dd></dl> where <span class="texhtml mvar" style="font-style:italic;">σ</span> denotes the sum-of-divisors function (i.e., the sum of all positive divisors of <span class="texhtml mvar" style="font-style:italic;">n</span>, including <span class="texhtml mvar" style="font-style:italic;">n</span> itself). The first few superabundant numbers are <span class="nowrap">1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ...</span> (sequence A004394 in the OEIS). For example, the number 5 is not a superabundant number because for <span class="nowrap">1, 2, 3, 4</span>, and 5, the sigma is <span class="nowrap">1, 3, 4, 7, 6</span>, and <span class="nowrap">7/4 &gt; 6/5</span>. Superabundant numbers were defined by Leonidas Alaoglu and Paul Erdős (1944). Unknown to Alaoglu and Erdős, about 30 pages of Ramanujan's 1915 paper "Highly Composite Numbers" were suppressed.  Those pages were finally published in <i>The Ramanujan Journal</i> 1 (1997), 119–153.  In section 59 of that paper, Ramanujan defines generalized highly composite numbers, which include the superabundant numbers. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Superabundant_number">https://en.wikipedia.org/wiki/Superabundant_number</a>)"""@en, """En mathématiques, un <b>nombre superabondant</b> est un entier naturel <i>n</i> tel que, pour tout <i>m</i> &lt; <i>n</i>,  <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 {\\rac {\\\\sigma (m)}{m}}<{\\rac {\\\\sigma (n)}{n}},}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mi>σ<!-- σ --></mi>               <mo stretchy="false">(</mo>               <mi>m</mi>               <mo stretchy="false">)</mo>             </mrow>             <mi>m</mi>           </mfrac>         </mrow>         <mo>&lt;</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mi>σ<!-- σ --></mi>               <mo stretchy="false">(</mo>               <mi>n</mi>               <mo stretchy="false">)</mo>             </mrow>             <mi>n</mi>           </mfrac>         </mrow>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\rac {\\\\sigma (m)}{m}}&lt;{\\rac {\\\\sigma (n)}{n}},}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c0c95524a4c441f12af88f13215170b97e547d4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.131ex; height:5.676ex;" alt="{\\\\displaystyle {\\rac {\\\\sigma (m)}{m}}<{\\rac {\\\\sigma (n)}{n}},}"></span></dd></dl> où <i>σ</i> est la fonction somme des diviseurs</span>. Les premiers nombres superabondants sont 1, 2, 4, 6, 12, 24, 36, 48, 60, 120 (suite A004394 de l'OEIS). Ce concept a été défini en 1944 par Leonidas Alaoglu et Paul Erdős</span>. Ces derniers ne savaient pas qu'en 1915, une trentaine de pages de l'article de Ramanujan <i>Highly composite numbers</i> (« Nombres hautement composés ») avaient été supprimées. Ces écrits furent finalement publiés en 1997, dans <i>The Ramanujan Journal 1</i>, <abbr class="abbr" title="page(s)">p.</abbr> 119-153. Dans la section <abbr class="abbr" title="numéro">n<sup>o</abbr> 59 de cet article, Ramanujan définit les nombres hautement composés, parmi lesquels figurent les nombres superabondants. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_superabondant">https://fr.wikipedia.org/wiki/Nombre_superabondant</a>)"""@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Nombre_superabondant>, <https://en.wikipedia.org/wiki/Superabundant_number> ;
  dc:created "2023-07-26"^^xsd:date ;
  skos:broader psr:-FM1M1PDT-5, psr:-CVDPQB0Q-M ;
  skos:inScheme psr: ;
  skos:prefLabel "superabundant number"@en, "nombre superabondant"@fr ;
  dc:modified "2024-10-18"^^xsd:date ;
  a skos:Concept .

psr:-FM1M1PDT-5
  skos:prefLabel "suite d'entiers"@fr, "integer sequence"@en ;
  a skos:Concept ;
  skos:narrower psr:-V2HDFL9H-Z .