Difference between revisions of "Template:Classes of natural numbers"

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{{Navbox
+
{{Navbox with collapsible groups
 
| name  = Classes of natural numbers
 
| name  = Classes of natural numbers
 
| title  = Classes of [[natural number]]s
 
| title  = Classes of [[natural number]]s
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| state  = {{{state|<includeonly>collapsed</includeonly>}}}
 
| state  = {{{state|<includeonly>collapsed</includeonly>}}}
  
|group1 = [[Exponentiation|Powers]] and<br/> related numbers
+
|group1 = [[Exponentiation|Powers]] and related numbers
 
|list1 =  
 
|list1 =  
 
* [[Achilles number|Achilles]]
 
* [[Achilles number|Achilles]]
 
* [[Power of two|Power of 2]]
 
* [[Power of two|Power of 2]]
 +
* [[Power of three|Power of 3]]
 
* [[Power of 10]]
 
* [[Power of 10]]
 
* [[Square number|Square]]
 
* [[Square number|Square]]
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* [[Sixth power]]
 
* [[Sixth power]]
 
* [[Seventh power]]
 
* [[Seventh power]]
 +
* [[Eighth power]]
 
* [[Perfect power]]
 
* [[Perfect power]]
 
* [[Powerful number|Powerful]]
 
* [[Powerful number|Powerful]]
 
* [[Prime power]]
 
* [[Prime power]]
  
|group2 = Of the form<br/> ''a'' &times; 2<sup>''b''</sup> ± 1
+
|group2 = Of the form ''a'' &times; 2<sup>''b''</sup> ± 1
 
|list2 =  
 
|list2 =  
 
* [[Cullen number|Cullen]]
 
* [[Cullen number|Cullen]]
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* [[Woodall number|Woodall]]
 
* [[Woodall number|Woodall]]
  
|group3 = Other polynomial<br/> numbers
+
|group3 = Other polynomial numbers
 
|list3 =  
 
|list3 =  
 
* [[Carol number|Carol]]
 
* [[Carol number|Carol]]
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* [[Loeschian number|Loeschian]]
 
* [[Loeschian number|Loeschian]]
 
* [[Lucky numbers of Euler]]
 
* [[Lucky numbers of Euler]]
* [[Repunit]]
 
  
|group4 = [[Recursion|Recursively]] defined<br/> numbers
+
|group4 = [[Recursion|Recursively]] defined numbers
 
|list4 =  
 
|list4 =  
 
* [[Fibonacci number|Fibonacci]]
 
* [[Fibonacci number|Fibonacci]]
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* [[Pell number|Pell]]
 
* [[Pell number|Pell]]
 
* [[Perrin number|Perrin]]
 
* [[Perrin number|Perrin]]
|group5 = Possessing a<br/> specific set<br/> of other numbers
+
|group5 = Possessing a specific set of other numbers
 
|list5 =  
 
|list5 =  
 
* [[Knödel number|Knödel]]
 
* [[Knödel number|Knödel]]
 
* [[Riesel number|Riesel]]
 
* [[Riesel number|Riesel]]
* [[Sierpinski number|Sierpinski]]
+
* [[Sierpiński number|Sierpiński]]
  
|group6 = Expressible via<br/> specific sums
+
|group6 = Expressible via specific sums
 
|list6 =  
 
|list6 =  
 
* [[Nonhypotenuse number|Nonhypotenuse]]
 
* [[Nonhypotenuse number|Nonhypotenuse]]
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   | list2 =  
 
   | list2 =  
 
* [[Tetrahedral number|Tetrahedral]]
 
* [[Tetrahedral number|Tetrahedral]]
 +
* [[Cube (algebra)|Cubic]]
 
* [[Octahedral number|Octahedral]]
 
* [[Octahedral number|Octahedral]]
 
* [[Dodecahedral number|Dodecahedral]]
 
* [[Dodecahedral number|Dodecahedral]]
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   |group1 = centered
 
   |group1 = centered
 
   |list1 =  
 
   |list1 =  
* [[Centered pentachoric number|Centered pentachoric]]
 
* [[Squared triangular number|Squared triangular]]
 
 
 
   |group2 = non-centered
 
   |group2 = non-centered
 
   |list2 =  
 
   |list2 =  
 
* [[Pentatope number|Pentatope]]
 
* [[Pentatope number|Pentatope]]
 +
* [[Squared triangular number|Squared triangular]]
 +
* [[Fourth power|Tesseractic]]
 
}}
 
}}
 
}}
 
}}
  
|group8 = [[Pseudoprime]]s
+
|group8 = Combinatorial numbers
 
|list8 =  
 
|list8 =  
* [[Carmichael number]]
 
* [[Catalan pseudoprime]]
 
* [[Elliptic pseudoprime]]
 
* [[Euler pseudoprime]]
 
* [[Euler–Jacobi pseudoprime]]
 
* [[Fermat pseudoprime]]
 
* [[Frobenius pseudoprime]]
 
* [[Lucas pseudoprime]]
 
* [[Somer–Lucas pseudoprime]]
 
* [[Strong pseudoprime]]
 
 
|group9 = Combinatorial<br/> numbers
 
|list9 =
 
 
* [[Bell number|Bell]]
 
* [[Bell number|Bell]]
 
* [[Cake number|Cake]]
 
* [[Cake number|Cake]]
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* [[Schröder–Hipparchus number|Schröder–Hipparchus]]
 
* [[Schröder–Hipparchus number|Schröder–Hipparchus]]
  
|group10 = [[Arithmetic function]]s
+
|group9 = [[Prime number|Primes]]
|list10 = {{Navbox|subgroup
+
|list9 =
  |group1 = By properties of [[Divisor function|σ(''n'')]]
+
* [[Wieferich prime#Wieferich numbers|Wieferich]]
 +
* [[Wall–Sun–Sun prime|Wall–Sun–Sun]]
 +
* [[Wolstenholme prime]]
 +
* [[Wilson prime#Wilson numbers|Wilson]]
 +
 
 +
|group10 = [[Pseudoprime]]s
 +
|list10 =
 +
* [[Carmichael number]]
 +
* [[Catalan pseudoprime]]
 +
* [[Elliptic pseudoprime]]
 +
* [[Euler pseudoprime]]
 +
* [[Euler–Jacobi pseudoprime]]
 +
* [[Fermat pseudoprime]]
 +
* [[Frobenius pseudoprime]]
 +
* [[Lucas pseudoprime]]
 +
* [[Lucas–Carmichael number]]
 +
* [[Somer–Lucas pseudoprime]]
 +
* [[Strong pseudoprime]]
 +
 
 +
|group11 = [[Arithmetic function]]s and [[Arithmetic dynamics|dynamics]]
 +
|list11 = {{Navbox|subgroup
 +
  |group1 = [[Divisor function]]s
 
  |list1 =  
 
  |list1 =  
 
* [[Abundant number|Abundant]]
 
* [[Abundant number|Abundant]]
 +
* [[Almost perfect number|Almost perfect]]
 
* [[Arithmetic number|Arithmetic]]
 
* [[Arithmetic number|Arithmetic]]
 +
* [[Betrothed numbers|Betrothed]]
 
* [[Colossally abundant number|Colossally abundant]]
 
* [[Colossally abundant number|Colossally abundant]]
 +
* [[Deficient number|Deficient]]
 
* [[Descartes number|Descartes]]
 
* [[Descartes number|Descartes]]
 
* [[Hemiperfect number|Hemiperfect]]
 
* [[Hemiperfect number|Hemiperfect]]
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* [[Multiply perfect number|Multiply perfect]]
 
* [[Multiply perfect number|Multiply perfect]]
 
* [[Perfect number|Perfect]]
 
* [[Perfect number|Perfect]]
* [[Practical number]]
+
* [[Practical number|Practical]]
 
* [[Primitive abundant number|Primitive abundant]]
 
* [[Primitive abundant number|Primitive abundant]]
 
* [[Quasiperfect number|Quasiperfect]]
 
* [[Quasiperfect number|Quasiperfect]]
 
* [[Refactorable number|Refactorable]]
 
* [[Refactorable number|Refactorable]]
 +
* [[Semiperfect number|Semiperfect]]
 
* [[Sublime number|Sublime]]
 
* [[Sublime number|Sublime]]
 
* [[Superabundant number|Superabundant]]
 
* [[Superabundant number|Superabundant]]
 
* [[Superior highly composite number|Superior highly composite]]
 
* [[Superior highly composite number|Superior highly composite]]
 
* [[Superperfect number|Superperfect]]
 
* [[Superperfect number|Superperfect]]
* [[Untouchable number|Untouchable]]
 
 
|group2 = By properties of [[Arithmetic function#Ω(n), ω(n), νp(n) – prime power decomposition|Ω(''n'')]]
 
|list2 =
 
* [[Almost prime]]
 
* [[Semiprime]]
 
  
  |group3 = By properties of [[Euler's totient function|φ(''n'')]]
+
  |group3 = [[Euler's totient function]]
 
  |list3 =  
 
  |list3 =  
 
* [[Highly cototient number|Highly cototient]]
 
* [[Highly cototient number|Highly cototient]]
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* [[Sparsely totient number|Sparsely totient]]
 
* [[Sparsely totient number|Sparsely totient]]
  
  |group4 = By properties of [[Divisor function#Definition|s(''n'')]]
+
  |group2 = [[Prime omega function]]s
  |list4 =  
+
  |list2 =
* [[Almost perfect number|Almost perfect]]
+
* [[Almost prime]]
 +
* [[Semiprime]]
 +
 
 +
|group5 = [[Aliquot sequence]]s
 +
|list5 =
 
* [[Amicable numbers|Amicable]]
 
* [[Amicable numbers|Amicable]]
* [[Betrothed numbers|Betrothed]]
 
* [[Deficient number|Deficient]]
 
 
* [[Perfect number|Perfect]]
 
* [[Perfect number|Perfect]]
* [[Semiperfect number|Semiperfect]]
 
 
* [[Sociable numbers|Sociable]]
 
* [[Sociable numbers|Sociable]]
 +
* [[Untouchable number|Untouchable]]
  
  |group5 = Other
+
  |group6 = [[Primorial]]
  |list5 =  
+
  |list6 =  
 
* [[Euclid number|Euclid]]
 
* [[Euclid number|Euclid]]
 
* [[Fortunate number|Fortunate]]
 
* [[Fortunate number|Fortunate]]
 
}}
 
}}
  
|group11 = Dividing a quotient
+
|group12 = Other [[prime factor]] or [[divisor]] related numbers
|list11 =
 
* [[Wieferich prime#Wieferich numbers|Wieferich]]
 
* [[Wall–Sun–Sun prime|Wall–Sun–Sun]]
 
* [[Wolstenholme prime]]
 
* [[Wilson prime#Wilson numbers|Wilson]]
 
 
 
|group12 = Other [[prime factor]]<br/> or [[divisor]] related<br/> numbers
 
 
|list12 =  
 
|list12 =  
 
* [[Blum integer|Blum]]
 
* [[Blum integer|Blum]]
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* [[Rough number|Rough]]
 
* [[Rough number|Rough]]
 
* [[Smooth number|Smooth]]
 
* [[Smooth number|Smooth]]
* [[Sociable number|Sociable]]
 
 
* [[Sphenic number|Sphenic]]
 
* [[Sphenic number|Sphenic]]
 
* [[Størmer number|Størmer]]
 
* [[Størmer number|Størmer]]
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* [[Zeisel number|Zeisel]]
 
* [[Zeisel number|Zeisel]]
  
|group13 = [[Number base|Base]]-dependent<br/> numbers
+
|group13 = [[Numeral system]]-dependent numbers
 
|list13 = {{Navbox|subgroup
 
|list13 = {{Navbox|subgroup
  |group1 = [[Numerical digit|Digit]]-based [[arithmetic dynamics]]
+
  |group1 = [[Arithmetic function]]s and [[Arithmetic dynamics|dynamics]]
 
  |list1 =  
 
  |list1 =  
 +
* [[Persistence of a number|Persistence]]
 +
** [[Additive persistence|Additive]]
 +
** [[Multiplicative persistence|Multiplicative]]
 +
{{Navbox|subgroup
 +
  |group1 = [[Digit sum]]
 +
  |list1 =
 
* [[Digit sum]]
 
* [[Digit sum]]
 
* [[Digital root]]
 
* [[Digital root]]
** [[Multiplicative digital root|Multiplicative]]
+
* [[Self number|Self]]
 +
* [[Sum-product number|Sum-product]]
 +
 
 +
  |group2 = Digit product
 +
  |list2 =
 +
* [[Multiplicative digital root]]
 +
* [[Sum-product number|Sum-product]]
 +
 
 +
  |group3 = Coding-related
 +
  |list3 =
 +
* [[Meertens number|Meertens]]
 +
 
 +
  |group4 = Other
 +
  |list4 =
 
* [[Dudeney number|Dudeney]]
 
* [[Dudeney number|Dudeney]]
 
* [[Factorion]]
 
* [[Factorion]]
* [[Happy number|Happy]]
+
* [[Kaprekar number|Kaprekar]]
 +
* [[Kaprekar's routine|Kaprekar's constant]]
 +
* [[Keith number|Keith]]
 
* [[Lychrel number|Lychrel]]
 
* [[Lychrel number|Lychrel]]
* [[Meertens number|Meertens]]
 
* [[Missing-digit sum]]
 
 
* [[Narcissistic number|Narcissistic]]
 
* [[Narcissistic number|Narcissistic]]
 
* [[Perfect digit-to-digit invariant]]
 
* [[Perfect digit-to-digit invariant]]
 
* [[Perfect digital invariant]]
 
* [[Perfect digital invariant]]
* [[Persistence of a number|Persistence]]
+
** [[Happy number|Happy]]
** [[Additive persistence|Additive]]
+
}}
** [[Multiplicative persistence|Multiplicative]]
+
|group2 = [[P-adic numbers]]-related
* [[Self number|Self]]
+
|list2 =
* [[Sum-product number|Sum-product]]
+
* [[Automorphic number|Automorphic]]
 +
** [[Trimorphic number|Trimorphic]]
  
  |group2 = Digit-composition related
+
  |group3 = [[Numerical digit|Digit]]-composition related
  |list2 =  
+
  |list3 =  
 
* [[Palindromic number|Palindromic]]
 
* [[Palindromic number|Palindromic]]
 
* [[Pandigital number|Pandigital]]
 
* [[Pandigital number|Pandigital]]
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* [[Undulating number|Undulating]]
 
* [[Undulating number|Undulating]]
  
  |group3 = Digit-[[permutation]] related
+
  |group4 = Digit-[[permutation]] related
  |list3 =  
+
  |list4 =  
 
* [[Cyclic number|Cyclic]]
 
* [[Cyclic number|Cyclic]]
 
* [[Digit-reassembly number|Digit-reassembly]]
 
* [[Digit-reassembly number|Digit-reassembly]]
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* [[Transposable integer|Transposable]]
 
* [[Transposable integer|Transposable]]
  
  |group4 = Divisor-related
+
  |group5 = Divisor-related
  |list4 =  
+
  |list5 =  
* [[Automorphic number|Automorphic]]
 
** [[Trimorphic number|Trimorphic]]
 
 
* [[Equidigital number|Equidigital]]
 
* [[Equidigital number|Equidigital]]
 
* [[Extravagant number|Extravagant]]
 
* [[Extravagant number|Extravagant]]
 
* [[Frugal number|Frugal]]
 
* [[Frugal number|Frugal]]
 
* [[Harshad number|Harshad]]
 
* [[Harshad number|Harshad]]
* [[Pernicious number|Pernicious]]
 
 
* [[Polydivisible number|Polydivisible]]
 
* [[Polydivisible number|Polydivisible]]
 +
* [[Smith number|Smith]]
 
* [[Vampire number|Vampire]]  
 
* [[Vampire number|Vampire]]  
  
  |group5 = Other
+
  |group6 = Other
  |list5 =  
+
  |list6 =  
 
* [[Friedman number|Friedman]]
 
* [[Friedman number|Friedman]]
* [[Kaprekar number|Kaprekar]]
 
* [[Keith number|Keith]]
 
 
}}
 
}}
  
|group14 = Generated<br/> via a [[Sieve theory|sieve]]
+
|group14 = [[Binary number]]s
|list14 =  
+
|list14 =
 +
* [[Evil number|Evil]]
 +
* [[Odious number|Odious]]
 +
* [[Pernicious number|Pernicious]]
 +
 
 +
|group15 = Generated via a [[Sieve theory|sieve]]
 +
|list15 =  
 
* [[Lucky number|Lucky]]
 
* [[Lucky number|Lucky]]
 
* [[Generation of primes|Prime]]
 
* [[Generation of primes|Prime]]
  
|group16 = Other numbers
+
|group16 = [[Sorting algorithm|Sorting]] related
 
|list16 =  
 
|list16 =  
 +
* [[Pancake sorting|Pancake number]]
 +
* [[Sorting number]]
 +
 +
|group17 = [[Natural language]] related
 +
|list17 =
 
* [[Aronson's sequence]]
 
* [[Aronson's sequence]]
 
* [[Ban number|Ban]]
 
* [[Ban number|Ban]]
 +
 +
|group18 = [[Graphemics]] related
 +
|list18 =
 
* [[Strobogrammatic number|Strobogrammatic]]
 
* [[Strobogrammatic number|Strobogrammatic]]
* [[Pancake sorting|Pancake number]]
 
  
 
| belowclass = hlist
 
| belowclass = hlist
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==See also==
 
==See also==
 
*{{tl|Divisor classes}}
 
*{{tl|Divisor classes}}
 +
*{{tl|Figurate numbers}}
 
*{{tl|Prime number classes}}
 
*{{tl|Prime number classes}}
 
*{{tl|Series (mathematics)}}
 
*{{tl|Series (mathematics)}}

Latest revision as of 15:52, 8 September 2020

Template documentation

Initial visibility: currently defaults to autocollapse

To set this template's initial visibility, the |state= parameter may be used:

  • |state=collapsed: {{Classes of natural numbers|state=collapsed}} to show the template collapsed, i.e., hidden apart from its title bar
  • |state=expanded: {{Classes of natural numbers|state=expanded}} to show the template expanded, i.e., fully visible
  • |state=autocollapse: {{Classes of natural numbers|state=autocollapse}}
    • shows the template collapsed to the title bar if there is a {{navbar}}, a {{sidebar}}, or some other table on the page with the collapsible attribute
    • shows the template in its expanded state if there are no other collapsible items on the page

If the |state= parameter in the template on this page is not set, the template's initial visibility is taken from the |default= parameter in the Collapsible option template. For the template on this page, that currently evaluates to autocollapse.

See also