@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:-FF87FBF1-H
  skos:related psr:-QBCKSDZ8-2 ;
  skos:inScheme psr: ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:definition """En arithmétique, un <b>nombre superparfait</b> est un entier strictement positif <span class="texhtml mvar" style="font-style:italic;">n</span> tel que  <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 \\\\sigma ^{2}(n)=\\\\sigma (\\\\sigma (n))=2n}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>σ<!-- σ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>σ<!-- σ --></mi>         <mo stretchy="false">(</mo>         <mi>σ<!-- σ --></mi>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mn>2</mn>         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sigma ^{2}(n)=\\\\sigma (\\\\sigma (n))=2n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/817124ff6a57093633fb75e306f3e608e5afd15c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.016ex; height:3.176ex;" alt="{\\\\displaystyle \\\\sigma ^{2}(n)=\\\\sigma (\\\\sigma (n))=2n}"></span>,</dd></dl> où <span class="texhtml mvar" style="font-style:italic;">σ</span> est la fonction somme des diviseurs. Les nombres superparfaits sont une généralisation des nombres parfaits. Le terme a été inventé par D. Suryanarayana (1969)</span>. Les  premiers nombres superparfaits sont :  <dl><dd>2, 4, 16, 64, 4096, 65536, 262144, 1073741824, <abbr class="abbr" title="et cetera">etc.</abbr> suite A019279 de l'OEIS</dd></dl> Pour illustrer : on peut voir que 16 est un nombre superparfait car σ(16) = 1 + 2 + 4 + 8 + 16 = 31, et σ(31) = 1 + 31 = 32, donc σ(σ(16) ) = 32 = 2 × 16. Si <i>n</i> est un nombre superparfait <i>pair</i>, alors <i>n</i> doit être une puissance de 2, disons 2<sup><i>k</i></sup>, telle que le nombre de Mersenne 2 <sup><i>k</i>+1</sup> − 1 soit premier</span>. On ne sait pas s'il existe des nombres superparfaits impairs. Un nombre superparfait impair <i>n</i> devrait être un nombre carré tel que <i>n</i> ou σ(<i>n</i>) soit divisible par au moins trois nombres premiers distincts. Il n'y a pas de nombres superparfaits impairs en dessous de 7  × 10<sup>24</sup>. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_superparfait">https://fr.wikipedia.org/wiki/Nombre_superparfait</a>)"""@fr, """In number theory, a <b>superperfect number</b> is a positive integer <span class="texhtml mvar" style="font-style:italic;">n</span> that satisfies  <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 \\\\sigma ^{2}(n)=\\\\sigma (\\\\sigma (n))=2n\\\\,,}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>σ<!-- σ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>σ<!-- σ --></mi>         <mo stretchy="false">(</mo>         <mi>σ<!-- σ --></mi>         <mo stretchy="false">(</mo>         <mi>n</mi>         <mo stretchy="false">)</mo>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mn>2</mn>         <mi>n</mi>         <mspace width="thinmathspace"></mspace>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\sigma ^{2}(n)=\\\\sigma (\\\\sigma (n))=2n\\\\,,}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f22f2b733462b493e03a69977a9b0f49c8d378" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.049ex; height:3.176ex;" alt="\\\\sigma^2(n)=\\\\sigma(\\\\sigma(n))=2n\\\\, ,"></span></dd></dl> where <span class="texhtml mvar" style="font-style:italic;">σ</span> is the divisor summatory function. Superperfect numbers are not a generalization of perfect numbers, but have a common generalization.  The term was coined by D. Suryanarayana (1969). The first few superperfect numbers are :  <dl><dd>2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... (sequence A019279 in the OEIS).</dd></dl> To illustrate: it can be seen that 16 is a superperfect number as <span class="nowrap">σ(16) = 1 + 2 + 4 + 8 + 16 = 31</span>, and <span class="nowrap">σ(31) = 1 + 31 = 32</span>, thus <span class="nowrap">σ(σ(16)) = 32 = 2 × 16</span>. If <span class="texhtml mvar" style="font-style:italic;">n</span> is an <i>even</i> superperfect number, then <span class="texhtml mvar" style="font-style:italic;">n</span> must be a power of 2, <span class="texhtml">2<sup><i>k</i></sup></span>, such that <span class="texhtml">2<sup><i>k</i>+1</sup> − 1</span> is a Mersenne prime. It is not known whether there are any odd superperfect numbers. An odd superperfect number <span class="texhtml mvar" style="font-style:italic;">n</span> would have to be a square number such that either <span class="texhtml mvar" style="font-style:italic;">n</span> or <span class="texhtml"><i>σ</i>(<i>n</i>)</span> is divisible by at least three distinct primes. There are no odd superperfect numbers below 7<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap"><span data-sort-value="7001240000000000000♠"></span>24</span></sup>.  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Superperfect_number">https://en.wikipedia.org/wiki/Superperfect_number</a>)"""@en ;
  skos:broader psr:-CVDPQB0Q-M, psr:-FM1M1PDT-5 ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Nombre_superparfait>, <https://en.wikipedia.org/wiki/Superperfect_number> ;
  skos:prefLabel "nombre superparfait"@fr, "superperfect number"@en ;
  a skos:Concept ;
  dc:created "2023-07-26"^^xsd:date .

psr:-CVDPQB0Q-M
  skos:prefLabel "natural numbers"@en, "entier naturel"@fr ;
  a skos:Concept ;
  skos:narrower psr:-FF87FBF1-H .

psr:-QBCKSDZ8-2
  skos:prefLabel "nombre parfait"@fr, "perfect number"@en ;
  a skos:Concept ;
  skos:related psr:-FF87FBF1-H .

psr: a skos:ConceptScheme .
psr:-FM1M1PDT-5
  skos:prefLabel "suite d'entiers"@fr, "integer sequence"@en ;
  a skos:Concept ;
  skos:narrower psr:-FF87FBF1-H .