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<title>Bellsche Zahl</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Bellsche Zahl</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Die Bellsche Zahl, Bellzahl oder Exponentialzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"> ist die Anzahl der <a href="Partition_(Mengenlehre)" title="Partition (Mengenlehre)">Partitionen</a> einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-elementigen Menge. Benannt ist sie nach dem Mathematiker <a href="Eric_Temple_Bell" class="mw-redirect" title="Eric Temple Bell">Eric Temple Bell</a>. Die Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{0},B_{1},B_{2},B_{3},\ldots }">

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<annotation encoding="application/x-tex">{\displaystyle B_{0},B_{1},B_{2},B_{3},\ldots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22ecd6eb6926cacaf172153c4c09ebc153b7c79c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.132ex; height:2.509ex;" alt="{\displaystyle B_{0},B_{1},B_{2},B_{3},\ldots }" loading="lazy"> beginnt mit

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<annotation encoding="application/x-tex">{\displaystyle 1,1,2,5,15,52,203,877,4140,21147,115975,678570,\ldots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb6a3a62cd6e18e0505863ee20c8726c0776bd3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:55.816ex; height:2.509ex;" alt="{\displaystyle 1,1,2,5,15,52,203,877,4140,21147,115975,678570,\ldots }" loading="lazy"> (Folge <a href="https://oeis.org/A000110" class="extiw external" title="oeis:A000110">A000110</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Bedeutung">Bedeutung</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Partitionen">Partitionen</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Partition_(Mengenlehre)" title="Partition (Mengenlehre)">Partition (Mengenlehre)</a></div>

Eine Partition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> einer Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> ist eine Menge <a href="Nichtleere_Menge" class="mw-redirect" title="Nichtleere Menge">nichtleerer</a>, <a href="Paarweise_disjunkt" class="mw-redirect" title="Paarweise disjunkt">paarweise disjunkter</a> <a href="Teilmenge" title="Teilmenge">Teilmengen</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, sodass jedes Element aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> in genau einer Menge aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> vorkommt. Für alle <a href="Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlichen Zahlen</a> einschließlich der Null <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cf5c8ad993619a325ca57a25c22cdc75a460f88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.968ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {N} _{0}}" loading="lazy"> bezeichnet nun die Bellsche Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"> die Anzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|Q\right|}">

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<annotation encoding="application/x-tex">{\displaystyle \left|Q\right|}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c2b0291540b8cae77b6b0821618b0dd79adc0d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.132ex; height:2.843ex;" alt="{\displaystyle \left|Q\right|}" loading="lazy"> der möglichen verschiedenen Partitionen einer Menge mit der <a href="M%C3%A4chtigkeit_(Mathematik)" title="Mächtigkeit (Mathematik)">Mächtigkeit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"> die Menge aller möglichen Partitionen darstellt. Formal:

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<annotation encoding="application/x-tex">{\displaystyle \left|M\right|=n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00ca0379e3aac37e53f6814311b9bdd51936fe97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.229ex; height:2.843ex;" alt="{\displaystyle \left|M\right|=n}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \forall P\in Q:\bigcup _{a\in P}a=M}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2e82671fc7e8c0458d07e78a04fb94624b70b80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.012ex; height:5.676ex;" alt="{\displaystyle \forall P\in Q:\bigcup _{a\in P}a=M}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \forall P\in Q:\forall a\in P:a\neq \emptyset }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd085a6fbfac76b200545c95a4b826e7f9e46b6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.19ex; height:2.843ex;" alt="{\displaystyle \forall P\in Q:\forall a\in P:a\neq \emptyset }" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \forall P\in Q:\forall a,b\in P:(a\neq b\Longrightarrow a\cap b=\emptyset )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4d98e1c6e8cafe0e5f2c3797b7bc08e45d5695c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.033ex; height:2.843ex;" alt="{\displaystyle \forall P\in Q:\forall a,b\in P:(a\neq b\Longrightarrow a\cap b=\emptyset )}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle B_{n}:=\left|Q\right|}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63baa21111825859d21bf89f776d0c75fdbb310f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.86ex; height:2.843ex;" alt="{\displaystyle B_{n}:=\left|Q\right|}" loading="lazy"></dd></dl>

Die Bellsche Zahl mit dem Index 0, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle B_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03a3f39a56ba486e7c6ec89b99f5ae2a21fa75b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle B_{0}}" loading="lazy">, – also die Anzahl der Partitionen der <a href="Leere_Menge" title="Leere Menge">leeren Menge</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }">

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<annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6af50205f42bb2ec3c666b7b847d2c7f96e464c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }" loading="lazy"> – ist 1, weil die einzige Partition der leeren Menge wieder die leere Menge selbst ist, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(\emptyset )=\{P_{1}\}=\{\emptyset \}}">

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<annotation encoding="application/x-tex">{\displaystyle Q(\emptyset )=\{P_{1}\}=\{\emptyset \}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d648e8eeae0b6110d263c19e8ded0d406f30202f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.366ex; height:2.843ex;" alt="{\displaystyle Q(\emptyset )=\{P_{1}\}=\{\emptyset \}}" loading="lazy">. Dies ist so, weil alle <a href="Aussage_(Logik)" title="Aussage (Logik)">Aussagen</a> mit dem <a href="Allquantor" class="mw-redirect" title="Allquantor">Allquantor</a> über die Elemente der leeren Menge wahr sind (siehe <a href="Leere_Menge" title="Leere Menge">leere Menge</a>).

<div class="mw-heading mw-heading3"><h3 id="Multiplikative_Partitionen">Multiplikative Partitionen</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Multiplikative_Partition" title="Multiplikative Partition">Multiplikative Partition</a></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> eine <a href="Quadratfreie_Zahl" title="Quadratfreie Zahl">quadratfreie Zahl</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega :\mathbb {N} \rightarrow \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle \omega :\mathbb {N} \rightarrow \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e0cf19176fd8a6787177f701e0fd36573102fa5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.353ex; height:2.176ex;" alt="{\displaystyle \omega :\mathbb {N} \rightarrow \mathbb {N} }" loading="lazy"> die Funktion zur Bestimmung der Anzahl der einzigartigen <a href="Primfaktoren" class="mw-redirect" title="Primfaktoren">Primfaktoren</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\omega (N)}">

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<annotation encoding="application/x-tex">{\displaystyle n=\omega (N)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a1c4da3d4349726df077f57e0ca6962aed5371d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.812ex; height:2.843ex;" alt="{\displaystyle n=\omega (N)}" loading="lazy">.

Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"> die Anzahl der unterschiedlichen <a href="Multiplikative_Partition" title="Multiplikative Partition">multiplikativen Partitionen</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">.

Sei zum Beispiel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=30}">

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<annotation encoding="application/x-tex">{\displaystyle N=30}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cae4bd0e7179a47f24bda656d5f5969c9d92507.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.487ex; height:2.176ex;" alt="{\displaystyle N=30}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\omega (30)=3}">

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<annotation encoding="application/x-tex">{\displaystyle n=\omega (30)=3}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/902ccc601d44b2896211d844713eb20f7b733bf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.334ex; height:2.843ex;" alt="{\displaystyle n=\omega (30)=3}" loading="lazy"> (da 30 aus den drei Primfaktoren 2, 3 und 5 besteht) und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{3}=5}">

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<annotation encoding="application/x-tex">{\displaystyle B_{3}=5}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/769c872a45e023a361982e572a3c8df98a526427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.079ex; height:2.509ex;" alt="{\displaystyle B_{3}=5}" loading="lazy"> ist damit die Anzahl der multiplikativen Partitionen. Diese lauten:

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<annotation encoding="application/x-tex">{\displaystyle 30=2\times 15=3\times 10=5\times 6=2\times 3\times 5}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24c9732bf20ddd721cb7aa4b399c718001ae0b18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:41.708ex; height:2.176ex;" alt="{\displaystyle 30=2\times 15=3\times 10=5\times 6=2\times 3\times 5}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3></div>

Für die Bellschen Zahlen gilt diese <a href="Rekursion" title="Rekursion">Rekursionsformel</a>:

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<annotation encoding="application/x-tex">{\displaystyle B_{n+1}=\sum _{k=0}^{n}{{n \choose k}B_{k}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eeb6d5e04444e1cd6822f84538219bad19b790aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.592ex; height:7.009ex;" alt="{\displaystyle B_{n+1}=\sum _{k=0}^{n}{{n \choose k}B_{k}}}" loading="lazy"></dd></dl>

Die Dobińskische Formel (Dobiński 1877)<a href="#cite_note-1">[1]</a>

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</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58b38f38ae808163c71e0587d51dc6bbd88221fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.475ex; height:7.009ex;" alt="{\displaystyle B_{n}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}}}" loading="lazy"></dd></dl>

erlaubt die explizite Definition der Bellschen Zahlen für alle n ≥ 0. Sie wurde nach dem polnischen Mathematiker Donald Gabriel Dobiński<a href="#cite_note-2">[2]</a> benannt.

Ihre Äquivalenz zur obigen Rekursionsformel lässt sich durch <a href="Vollst%C3%A4ndige_Induktion" title="Vollständige Induktion">vollständige Induktion</a> beweisen:

Sei

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</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(n)={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cf7b8121acbe9fd91f0fbc18227c6a1973d2189.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.975ex; height:7.009ex;" alt="{\displaystyle f(n)={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}}}" loading="lazy"> .</dd></dl>

Dann gilt:

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</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(0)={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!}}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15982c73a690afe7938d22ac7dff62f7bd3dfc1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.432ex; height:7.009ex;" alt="{\displaystyle f(0)={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!}}=1}" loading="lazy"></dd></dl>

sowie für n ≥ 0:

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</msup>

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</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

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</mrow>

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</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

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</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(n+1)={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n+1}}{k!}}={\frac {1}{e}}\sum _{k=1}^{\infty }{\frac {k^{n+1}}{k!}}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {(k+1)^{n+1}}{(k+1)!}}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {(k+1)^{n}}{k!}}=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab7a443c592be9047c99e75c8f2248ac6d0495c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:78.833ex; height:7.176ex;" alt="{\displaystyle f(n+1)={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n+1}}{k!}}={\frac {1}{e}}\sum _{k=1}^{\infty }{\frac {k^{n+1}}{k!}}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {(k+1)^{n+1}}{(k+1)!}}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {(k+1)^{n}}{k!}}=}" loading="lazy"></dd>

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</mrow>

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<mrow>

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</mrow>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!}}\sum _{m=0}^{n}{\binom {n}{m}}k^{m}=\sum _{m=0}^{n}{\binom {n}{m}}{\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{m}}{k!}}=\sum _{m=0}^{n}{\binom {n}{m}}f(m)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07802a1703fae220e2a8b297d079cd53b20f99db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:63.9ex; height:7.009ex;" alt="{\displaystyle ={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!}}\sum _{m=0}^{n}{\binom {n}{m}}k^{m}=\sum _{m=0}^{n}{\binom {n}{m}}{\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{m}}{k!}}=\sum _{m=0}^{n}{\binom {n}{m}}f(m)}" loading="lazy"></dd></dl>

Aus

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(0)=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00884d9c3ef0dbc18fe185a4444f1da931c8ba0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=1}" loading="lazy"> und</dd></dl>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall n\in \mathbb {N} _{0}:f(n+1)=\sum _{m=0}^{n}{\binom {n}{m}}f(m)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3074e3b86b1b98700cf1854866f07bd7694eaa0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.302ex; height:6.843ex;" alt="{\displaystyle \forall n\in \mathbb {N} _{0}:f(n+1)=\sum _{m=0}^{n}{\binom {n}{m}}f(m)}" loading="lazy">

folgt schließlich:

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall n\in \mathbb {N} _{0}:f(n)=B_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/165cec110777279529ed7a4610426a9b93f51d15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.761ex; height:2.843ex;" alt="{\displaystyle \forall n\in \mathbb {N} _{0}:f(n)=B_{n}}" loading="lazy"></dd></dl>

Somit ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"> auch das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-te <a href="Moment_(Stochastik)" title="Moment (Stochastik)">Moment</a> einer <a href="Poisson-Verteilung" title="Poisson-Verteilung">Poisson-Verteilung</a> mit <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a> 1.

<div class="mw-heading mw-heading3"><h3 id="Erzeugende_Funktionen">Erzeugende Funktionen</h3></div>

Die <a href="Erzeugende_Funktion" title="Erzeugende Funktion">erzeugende Funktion</a> der Bellzahlen ist wie folgt darstellbar:

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

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</msup>

<mfrac>

</mfrac>

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</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

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</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }B_{n}\,x^{n}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!\,(1-kx)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9ff5c15efe944c466c29624551d6df762621882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.32ex; height:7.009ex;" alt="{\displaystyle \sum _{n=0}^{\infty }B_{n}\,x^{n}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!\,(1-kx)}}}" loading="lazy"></dd></dl>

Die exponentiell erzeugende Funktion lautet:

</mrow>

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</msup>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}\,x^{n}=e^{e^{x}-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cac9ef9c564bc9358d7d3a3e7b26afca8bcba3de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.704ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}\,x^{n}=e^{e^{x}-1}}" loading="lazy"></dd></dl>

Dies folgt aus der genannten Dobiński-Formel:

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</munderover>

<mfrac>

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<mfrac>

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</munderover>

<mfrac>

<msup>

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</msup>

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</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

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</munderover>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

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</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

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</mrow>

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</munderover>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

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</msup>

<mrow>

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</mfrac>

</mrow>

<msup>

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</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

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</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}x^{n}=\sum _{n=0}^{\infty }{\frac {1}{n!}}{\biggl [}{\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}}{\biggr ]}x^{n}={\frac {1}{e}}\sum _{n=0}^{\infty }\sum _{k=0}^{\infty }{\frac {k^{n}}{k!n!}}x^{n}={\frac {1}{e}}\sum _{k=0}^{\infty }\sum _{n=0}^{\infty }{\frac {k^{n}}{k!n!}}x^{n}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!}}\sum _{n=0}^{\infty }{\frac {(kx)^{n}}{n!}}=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/957d4dbce9d12ee2de2f4276b8207f3d17738ea5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:99.636ex; height:7.009ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}x^{n}=\sum _{n=0}^{\infty }{\frac {1}{n!}}{\biggl [}{\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {k^{n}}{k!}}{\biggr ]}x^{n}={\frac {1}{e}}\sum _{n=0}^{\infty }\sum _{k=0}^{\infty }{\frac {k^{n}}{k!n!}}x^{n}={\frac {1}{e}}\sum _{k=0}^{\infty }\sum _{n=0}^{\infty }{\frac {k^{n}}{k!n!}}x^{n}={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!}}\sum _{n=0}^{\infty }{\frac {(kx)^{n}}{n!}}=}" loading="lazy"></dd>

<mfrac>

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</munderover>

<mfrac>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!}}\exp(kx)={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!}}\exp(x)^{k}={\frac {1}{e}}\exp[\exp(x)]=\exp[\exp(x)-1]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88441676e1fc8fd430451497994d921d9bf506cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:75.313ex; height:7.009ex;" alt="{\displaystyle ={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!}}\exp(kx)={\frac {1}{e}}\sum _{k=0}^{\infty }{\frac {1}{k!}}\exp(x)^{k}={\frac {1}{e}}\exp[\exp(x)]=\exp[\exp(x)-1]}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Kongruenzsätze">Kongruenzsätze</h3></div>

Die Bellschen Zahlen genügen der <a href="Kongruenz_(Zahlentheorie)" title="Kongruenz (Zahlentheorie)">Kongruenz</a> (Touchard 1933)<a href="#cite_note-3">[3]</a>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle B_{p^{k}+n}\equiv k\,B_{n}+B_{n+1}\ ({\text{mod }}p)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5b07cb41a0500291e4fc7f4b08d03c65adc22a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.081ex; height:3.176ex;" alt="{\displaystyle B_{p^{k}+n}\equiv k\,B_{n}+B_{n+1}\ ({\text{mod }}p)}" loading="lazy"></dd></dl>

für natürliche Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> und <a href="Primzahl" title="Primzahl">Primzahlen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, insbesondere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p^{k}}\equiv k+1\ ({\text{mod }}p)}">

<msub>

<msup>

</mrow>

</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{p^{k}}\equiv k+1\ ({\text{mod }}p)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ded909c4afd58179a66986c739a2cf3bb63702b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.526ex; height:3.176ex;" alt="{\displaystyle B_{p^{k}}\equiv k+1\ ({\text{mod }}p)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{p}\equiv 2\ ({\text{mod }}p)}">

<msub>

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</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{p}\equiv 2\ ({\text{mod }}p)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b93f845aab9e8129ef8f9b1b6273bb5f8d6c9043.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.615ex; height:3.009ex;" alt="{\displaystyle B_{p}\equiv 2\ ({\text{mod }}p)}" loading="lazy"> und, nach <a href="Iteration" title="Iteration">Iteration</a>,<a href="#cite_note-4">[4]</a>

<msub>

<msup>

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</msup>

</mrow>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle B_{1+p+\ldots +p^{p-1}+n}\equiv B_{n}\ ({\text{mod }}p).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c5ad235f23b821b3ab2c6891b0e69742a49dd81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.298ex; height:3.176ex;" alt="{\displaystyle B_{1+p+\ldots +p^{p-1}+n}\equiv B_{n}\ ({\text{mod }}p).}" loading="lazy"></dd></dl>

Es wird vermutet, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+p+\dots +p^{p-1}}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+p+\dots +p^{p-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba35600779b2810ebb18170fc488f39d3b61b92c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.905ex; height:3.009ex;" alt="{\displaystyle 1+p+\dots +p^{p-1}}" loading="lazy"> die kleinste <a href="Periodizit%C3%A4t_(Mathematik)" class="mw-redirect" title="Periodizität (Mathematik)">Periode</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}\ ({\text{mod }}p)}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}\ ({\text{mod }}p)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7554d1b1c3d4d81ff53d6c77cdef66e57e6a565c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.513ex; height:2.843ex;" alt="{\displaystyle B_{n}\ ({\text{mod }}p)}" loading="lazy"> ist.<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> Für Primzahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p>2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0502012bc3b4e73e6f3c2f4748feaab3fd3c350d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>2}" loading="lazy"> ist

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{p^{k+1}n}\equiv B_{p^{k}n+1}\ ({\text{mod }}p^{k+1}),}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/149a7b6c706a8b07378b072a6c9cce7c1f9b2fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.609ex; height:3.509ex;" alt="{\displaystyle B_{p^{k+1}n}\equiv B_{p^{k}n+1}\ ({\text{mod }}p^{k+1}),}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d62e4100b94c1939c67f2d4b8580d26c78106c44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=2}" loading="lazy"> gilt die Kongruenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\text{mod }}p^{k})}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\text{mod }}p^{k})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12b31e0b6392d6fdd3ec9f92cd50316bdb508152.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.039ex; height:3.176ex;" alt="{\displaystyle ({\text{mod }}p^{k})}" loading="lazy">.<a href="#cite_note-7">[7]</a>

Da die <a href="Stirling-Zahl" title="Stirling-Zahl">Stirling-Zahl</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(n,k)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(n,k)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e175d023088543773687c25fa875528f091ed34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.948ex; height:2.843ex;" alt="{\displaystyle S(n,k)}" loading="lazy"> zweiter Art die Anzahl der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-Partitionen einer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-elementigen Menge ist, gilt

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}=\sum _{k=0}^{n}S(n,k).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bae139ffa25497affa2f79f03025fdd9818812db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.418ex; height:7.009ex;" alt="{\displaystyle B_{n}=\sum _{k=0}^{n}S(n,k).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Asymptotik">Asymptotik</h2></div>

Für die Bellzahlen sind verschiedene asymptotische Formeln bekannt, etwa

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}\sim n^{-1/2}\ {\bigl (}\lambda (n){\bigr )}^{n+1/2}\ e^{\lambda (n)-n-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff3566dbf28c9fcf1e8c51e05f12c9925e1af0c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.171ex; height:3.843ex;" alt="{\displaystyle B_{n}\sim n^{-1/2}\ {\bigl (}\lambda (n){\bigr )}^{n+1/2}\ e^{\lambda (n)-n-1}}" loading="lazy">     mit     <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (n)=e^{W(n)}={\frac {n}{W(n)}}}">

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda (n)=e^{W(n)}={\frac {n}{W(n)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5abbdb167fbc37d38d58c54e2560cd041987d327.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.535ex; height:5.509ex;" alt="{\displaystyle \lambda (n)=e^{W(n)}={\frac {n}{W(n)}}}" loading="lazy"></dd></dl>

mit der <a href="Lambert-W-Funktion" class="mw-redirect" title="Lambert-W-Funktion">Lambert-W-Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Bellsches_Dreieck">Bellsches Dreieck</h2></div>

Die Bellschen Zahlen lassen sich intuitiv durch das Bellsche Dreieck erzeugen, welches – wie das <a href="Pascalsches_Dreieck" title="Pascalsches Dreieck">Pascalsche Dreieck</a> – aus natürlichen Zahlen besteht und pro Zeile ein Element bzw. eine Spalte mehr besitzt. Das Bellsche Dreieck wird gelegentlich auch Aitkens array (nach <a href="Alexander_Aitken" title="Alexander Aitken">Alexander Aitken</a>) oder Peirce-Dreieck (nach <a href="Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a>) genannt.

Es wird nach den folgenden Regeln konstruiert:

<ol><li>Die erste Zeile hat nur ein Element: Die Eins: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x_{1,1}=1\ }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ x_{1,1}=1\ }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b51a7339f42e551fbb0850da94d9909f2daaa2b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.086ex; height:2.843ex;" alt="{\displaystyle \ x_{1,1}=1\ }" loading="lazy">.</li>

<li>Jede folgende Zeile hat jeweils ein Element mehr als die vorherige Zeile, d. h. die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-te Zeile hat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> Elemente.</li>

<li>Das jeweils erste Element jeder Zeile hat den gleichen Wert wie das letzte Element der vorherigen Zeile: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x_{k+1,1}=x_{k,k}\ }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ x_{k+1,1}=x_{k,k}\ }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4100123224f54221848322f684d0556f3ef1440d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.79ex; height:2.343ex;" alt="{\displaystyle \ x_{k+1,1}=x_{k,k}\ }" loading="lazy">.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-te Elemente der n. Zeile (für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<k\leq n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1<k\leq n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad11f2a42f21642e5e85107acd2b36d7e51f3353.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.965ex; height:2.343ex;" alt="{\displaystyle 1<k\leq n}" loading="lazy">) ist gleich der Summe des links stehenden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k-1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k-1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e69f74fa2adbbab50f6969acb2af719045435461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k-1)}" loading="lazy">-ten Elements derselben Zeile und des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k-1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k-1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e69f74fa2adbbab50f6969acb2af719045435461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.023ex; height:2.843ex;" alt="{\displaystyle (k-1)}" loading="lazy">-ten Elements der vorherigen Zeile (also jene mit der Nummer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy">): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ x_{k,n}=x_{k,n-1}+x_{k-1,n-1}\ }">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ x_{k,n}=x_{k,n-1}+x_{k-1,n-1}\ }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dfcbf38e67da160371cfb742ea459a3217add0ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.988ex; height:2.676ex;" alt="{\displaystyle \ x_{k,n}=x_{k,n-1}+x_{k-1,n-1}\ }" loading="lazy">.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"> ist nun das <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-te Element aus der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-ten Zeile <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {x_{n,n}}}}">

<msub>

</mrow>

</msub>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {x_{n,n}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afa7f840272acd7a01f3701f8568a9f0c5aea066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.179ex; margin-bottom: -0.826ex; width:3.994ex; height:3.343ex;" alt="{\displaystyle {\underline {x_{n,n}}}}" loading="lazy"> (bzw. das erste Element aus der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy">-ten Zeile <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n+1,1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n+1,1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc59a180274dc28567f6fd9f794311fd92bcb52b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.928ex; height:2.343ex;" alt="{\displaystyle x_{n+1,1}}" loading="lazy">).</li></ol>

Die ersten sechs Zeilen, erzeugt nach diesen Regeln, sehen wie folgt aus:

<td style="background:#FFD0D0">1

</td></tr>

<td>2

</td></tr>

<td>5

</td></tr>

<td>15

</td></tr>

<td>52

</td></tr>

<td>203

</td></tr>

<tr>

<td>…

</td></tr></tbody></table></dd></dl>

Wegen des dritten Schritts sind die Bellschen Zahlen sowohl auf der linken als auch auf der rechten Kante des Dreiecks zu sehen, lediglich mit dem Unterschied, dass in der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-ten Zeile links die Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n-1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35a38fdd982d39522072e328364b4ccec132f6dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.083ex; height:2.509ex;" alt="{\displaystyle B_{n-1}}" loading="lazy"> und rechts die Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"> steht.

<div class="mw-heading mw-heading2"><h2 id="Bellsche_Primzahlen">Bellsche Primzahlen</h2></div>

Im Jahre <a href="1978" title="1978">1978</a> formulierte <a href="Martin_Gardner" title="Martin Gardner">Martin Gardner</a> die Frage, ob unendlich viele Bellsche Zahlen auch <a href="Primzahlen" class="mw-redirect" title="Primzahlen">Primzahlen</a> sind. Die ersten Bellschen Primzahlen sind:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> (Folge <a href="https://oeis.org/A051130" class="extiw external" title="oeis:A051130">A051130</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"> (Folge <a href="https://oeis.org/A051131" class="extiw external" title="oeis:A051131">A051131</a> in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)

</th></tr>

<tr>

<td>2

</td></tr>

<tr>

<td>5

</td></tr>

<tr>

<td>877

</td></tr>

<tr>

<td>27644437

</td></tr>

<tr>

<td>35742549198872617291353508656626642567

</td></tr>

<tr>

<td>359334085968622831041960188598043661065388726959079837

</td></tr></tbody></table>

Die nächste Bellsche Primzahl ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2841}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{2841}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be484403df4f7078f86011d25de2bd772379ffa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.284ex; height:2.509ex;" alt="{\displaystyle B_{2841}}" loading="lazy">, die etwa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9{,}30740105\times 10^{6538}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 9{,}30740105\times 10^{6538}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82569fd203d55ab4d2deaf127504452799c0f8fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.794ex; height:3.009ex;" alt="{\displaystyle 9{,}30740105\times 10^{6538}}" loading="lazy"> beträgt.<a href="#cite_note-8">[8]</a> Sie ist auch die aktuell größte bekannte Bellsche Primzahl (Stand: 5. August 2018). Im Jahre 2002 zeigte Phil Carmody zunächst, dass es sich bei dieser Zahl wahrscheinlich um eine Primzahl (eine sogenannte <a href="PRP-Zahl" title="PRP-Zahl">PRP-Zahl</a>) handelt, sie also entweder tatsächlich eine echte Primzahl oder eine <a href="Pseudoprimzahl" title="Pseudoprimzahl">Pseudoprimzahl</a> ist. Nach einer 17-monatigen Berechnung mit Marcel Martins Programm „Primo“, welches mit einem Verfahren mit <a href="Elliptische_Kurven" class="mw-redirect" title="Elliptische Kurven">elliptischen Kurven</a> arbeitet, bewies Ignacio Larrosa Cañestro schließlich im Jahre 2004, dass es sich bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{2841}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{2841}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be484403df4f7078f86011d25de2bd772379ffa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.284ex; height:2.509ex;" alt="{\displaystyle B_{2841}}" loading="lazy"> um eine Primzahl handelt. Gleichzeitig schloss er weitere Bellsche Primzahlen bis zu einer Grenze von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{6000}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{6000}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3e9b417694fe05791c3ca5f670f0768214160c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.284ex; height:2.509ex;" alt="{\displaystyle B_{6000}}" loading="lazy"> aus, welche später von <a href="Eric_Weisstein" title="Eric Weisstein">Eric Weisstein</a> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{30447}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{30447}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85342a6e881f8e7dba26cf3a5064394d658b4215.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.106ex; height:2.509ex;" alt="{\displaystyle B_{30447}}" loading="lazy"> angehoben wurde.

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> G. Dobiński: <a rel="nofollow" class="external text" href="http://www.archive.org/stream/archivdermathem88unkngoog#page/n346/mode/1up">Summirung der Reihe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum {\frac {n^{m}}{n!}}}">

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum {\frac {n^{m}}{n!}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94a1765ce698863884e04f46b7795cdfe605ed9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.999ex; height:3.843ex;" alt="{\displaystyle \textstyle \sum {\frac {n^{m}}{n!}}}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=1,2,3,4,5,\ldots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m=1,2,3,4,5,\ldots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b92c3dc7c6a039fffd618caf694ed6200489b42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.844ex; height:2.509ex;" alt="{\displaystyle m=1,2,3,4,5,\ldots }" loading="lazy"></a>, Grunert-Archiv 61, 1877, S. 333–336

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a rel="nofollow" class="external text" href="https://yyiki.org/wiki/Person/G.%20Dob%C3%ADnski/">YYiki: G. Dobínski.</a> Abgerufen am 7. September 2021.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3ABellsche+Zahl&rft.title=YYiki%3A+G.+Dob%C3%ADnski&rft.description=YYiki%3A+G.+Dob%C3%ADnski&rft.identifier=https%3A%2F%2Fyyiki.org%2Fwiki%2FPerson%2FG.%2520Dob%25C3%25ADnski%2F">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Jacques Touchard: Propriétés arithmétiques de certains nombres récurrents, Annales de la Société scientifique de Bruxelles A 53, 1933, S. 21–31 (französisch)

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> <a href="Marshall_Hall_(Mathematiker)" title="Marshall Hall (Mathematiker)">Marshall Hall</a>: <a rel="nofollow" class="external text" href="http://www.ams.org/journals/bull/1934-40-05/S0002-9904-1934-05853-1/home.html">Arithmetic properties of a partition function</a>, Bulletin of the AMS 40, 1934, S. 387 (englisch; nur Abstract)

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Christian Radoux: Nombres de Bell, modulo p premier, et extensions de degré p de Fp, Comptes rendus hebdomadaires des séances de l’académie des sciences 281 A, 1975, S. 879–882 (französisch)

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> <a href="Peter_Montgomery_(Mathematiker)" title="Peter Montgomery (Mathematiker)">Peter L. Montgomery</a>, Sangil Nahm, <a href="Samuel_Wagstaff" title="Samuel Wagstaff">Samuel S. Wagstaff</a>: <a rel="nofollow" class="external text" href="http://www.cs.purdue.edu/homes/ssw/mcom2340.pdf">The period of the Bell numbers modulo a prime</a> (<a href="PDF" class="mw-redirect" title="PDF">PDF</a>-Datei, 168 kB), Mathematics of computation 79, 2010, S. 1793–1800 (englisch)

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Anne Gertsch, <a href="Alain_Robert_(Mathematiker)" title="Alain Robert (Mathematiker)">Alain M. Robert</a>: <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.bbms/1105554416">Some congruences concerning the Bell numbers</a>, Bulletin of the Belgian Mathematical Society – Simon Stevin 3, 1996, S. 467–475 (englisch)

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> <a rel="nofollow" class="external text" href="http://primes.utm.edu/primes/page.php?id=68825">93074010508593618333...(6499 other digits)...83885253703080601131 auf Prime Pages</a>. Abgerufen am 5. August 2018.

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Eric_Temple_Bell" class="mw-redirect" title="Eric Temple Bell">Eric Temple Bell</a>: Exponential Numbers, The American Mathematical Monthly 41, 1934, S. 411–419</li>

<li>Jacques Touchard: <a rel="nofollow" class="external text" href="http://books.google.de/books?id=x34z99fCRbsC&pg=PA305">Nombres exponentiels et nombres de Bernoulli</a>, Canadian Journal of Mathematics 8, 1956, S. 305–320 (französisch)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/BellNumber.html">Bell Number</a> und <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/DobinskisFormula.html">Dobiński’s Formula</a>. In <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch)</li>

<li><a rel="nofollow" class="external text" href="http://functions.wolfram.com/IntegerFunctions/BellB/">Bell numbers</a> bei The Wolfram Functions Site (englisch; mit Berechnungsmöglichkeit)</li>

<li><a rel="nofollow" class="external text" href="http://dlmf.nist.gov/26.7">Set Partitions: Bell Numbers</a> in der NIST Digital Library of Mathematical Functions (englisch)</li>

<li>Peter Luschny: <a rel="nofollow" class="external text" href="http://oeis.org/wiki/User:Peter_Luschny/SetPartitions">Set partitions and Bell numbers</a> (englisch). Eine Zusammenfassung von OEIS-Folgen zu den Bellzahlen im <a href="OEIS" class="mw-redirect" title="OEIS">OEIS</a> Wiki.</li></ul></div><div><div class="zim-footer">

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