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

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(Mention squares (can't believe they were not here) and perfect powers)
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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
 
| listclass = hlist
 
| listclass = hlist
| state  = {{{state|collapsed}}}
+
| state  = {{{state|<includeonly>collapsed</includeonly>}}}
  
|group1 = [[Exponentiation|Powers]] and<br/> related numbers
+
|group1 = [[Exponentiation|Powers]] and related numbers
 
|list1 =  
 
|list1 =  
* [[Achilles number]]s
+
* [[Achilles number|Achilles]]
* [[Power of two|Powers of two]]
+
* [[Power of two|Power of 2]]
* [[Square number|Square]]s
+
* [[Power of three|Power of 3]]
* [[Cube (algebra)|Cubes]]
+
* [[Power of 10]]
* [[Fourth power]]s
+
* [[Square number|Square]]
* [[Fifth power (algebra)|Fifth powers]]
+
* [[Cube (algebra)|Cube]]
* [[Perfect power]]s
+
* [[Fourth power]]
* [[Powerful number]]s
+
* [[Fifth power (algebra)|Fifth power]]
 +
* [[Sixth power]]
 +
* [[Seventh power]]
 +
* [[Eighth power]]
 +
* [[Perfect power]]
 +
* [[Powerful number|Powerful]]
 +
* [[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]]s
+
* [[Cullen number|Cullen]]
* [[Double Mersenne number]]s
+
* [[Double Mersenne number|Double Mersenne]]
* [[Fermat number]]s
+
* [[Fermat number|Fermat]]
* [[Mersenne number]]s
+
* [[Mersenne prime|Mersenne]]
* [[Proth number]]s
+
* [[Proth number|Proth]]
* [[Thabit number]]s
+
* [[Thabit number|Thabit]]
* [[Woodall number]]s
+
* [[Woodall number|Woodall]]
  
|group3 = other polynomial<br/> numbers
+
|group3 = Other polynomial numbers
 
|list3 =  
 
|list3 =  
* [[Carol number]]s
+
* [[Carol number|Carol]]
* [[Hilbert number]]s
+
* [[Hilbert number|Hilbert]]
* [[Idoneal number]]s
+
* [[Idoneal number|Idoneal]]
* [[Kynea number]]s
+
* [[Kynea number|Kynea]]
* [[Leyland number]]s
+
* [[Leyland number|Leyland]]
 +
* [[Loeschian number|Loeschian]]
 
* [[Lucky numbers of Euler]]
 
* [[Lucky numbers of Euler]]
  
|group4 = [[Recursion|Recursively]] defined<br/> numbers
+
|group4 = [[Recursion|Recursively]] defined numbers
 
|list4 =  
 
|list4 =  
* [[Fibonacci number]]s
+
* [[Fibonacci number|Fibonacci]]
* [[Jacobsthal number]]s
+
* [[Jacobsthal number|Jacobsthal]]
* [[Leonardo number]]s
+
* [[Leonardo number|Leonardo]]
* [[Pell number]]s
+
* [[Lucas number|Lucas]]
* [[Perrin number]]s
+
* [[Padovan sequence|Padovan]]
 
+
* [[Pell number|Pell]]
|group5 = Possessing a<br/> specific set<br/> of other numbers
+
* [[Perrin number|Perrin]]
 +
|group5 = Possessing a specific set of other numbers
 
|list5 =  
 
|list5 =  
* [[Knödel number]]s
+
* [[Knödel number|Knödel]]
* [[Riesel number]]s
+
* [[Riesel number|Riesel]]
* [[Sierpinski number]]s
+
* [[Sierpiński number|Sierpiński]]
  
|group6 = Expressible via<br/> specific sums
+
|group6 = Expressible via specific sums
 
|list6 =  
 
|list6 =  
* [[Nonhypotenuse number]]s
+
* [[Nonhypotenuse number|Nonhypotenuse]]
* [[Polite number]]s
+
* [[Polite number|Polite]]
* [[Practical number]]s
+
* [[Practical number|Practical]]
* [[Primary pseudoperfect number]]s
+
* [[Primary pseudoperfect number|Primary pseudoperfect]]
* [[Ulam number]]s
+
* [[Ulam number|Ulam]]
* [[Wolstenholme number]]s
+
* [[Wolstenholme number|Wolstenholme]]
 
 
|group7 = Generated<br/> via a [[Sieve theory|sieve]]
 
|list7 =
 
* [[Lucky number]]s
 
 
 
|group8 = [[Code]] related
 
|list8 =
 
* [[Meertens number]]s
 
  
|group9 = [[Figurate number]]s
+
|group7 = [[Figurate number]]s
|list9 = {{Navbox subgroup
+
|list7 = {{Navbox|subgroup
 
  | group1 = [[Two-dimensional space|2-dimensional]]
 
  | group1 = [[Two-dimensional space|2-dimensional]]
  | list1 = {{Navbox subgroup
+
  | list1 = {{Navbox|subgroup
 
   | group1 = [[Centered polygonal number|centered]]
 
   | group1 = [[Centered polygonal number|centered]]
 
   | list1 =  
 
   | list1 =  
* [[Centered triangular number]]s
+
* [[Centered triangular number|Centered triangular]]
* [[Centered square number]]s
+
* [[Centered square number|Centered square]]
* [[Centered pentagonal number]]s
+
* [[Centered pentagonal number|Centered pentagonal]]
* [[Centered hexagonal number]]s
+
* [[Centered hexagonal number|Centered hexagonal]]
* [[Centered heptagonal number]]s
+
* [[Centered heptagonal number|Centered heptagonal]]
* [[Centered octagonal number]]s
+
* [[Centered octagonal number|Centered octagonal]]
* [[Centered nonagonal number]]s
+
* [[Centered nonagonal number|Centered nonagonal]]
* [[Centered decagonal number]]s
+
* [[Centered decagonal number|Centered decagonal]]
* [[Star number]]s
+
* [[Star number|Star]]
  
 
   | group2 = [[Polygonal number|non-centered]]
 
   | group2 = [[Polygonal number|non-centered]]
 
   | list2 =  
 
   | list2 =  
* [[Triangular number]]s
+
* [[Triangular number|Triangular]]
* [[Pentagonal number]]s
+
* [[Square number|Square]]
* [[Hexagonal number]]s
+
* [[Square triangular number|Square triangular]]
* [[Heptagonal number]]s
+
* [[Pentagonal number|Pentagonal]]
* [[Octagonal number]]s
+
* [[Hexagonal number|Hexagonal]]
* [[Nonagonal number]]s
+
* [[Heptagonal number|Heptagonal]]
* [[Decagonal number]]s
+
* [[Octagonal number|Octagonal]]
* [[Dodecagonal number]]s
+
* [[Nonagonal number|Nonagonal]]
 +
* [[Decagonal number|Decagonal]]
 +
* [[Dodecagonal number|Dodecagonal]]
 
}}
 
}}
  
 
  | group2 = [[Three-dimensional space|3-dimensional]]
 
  | group2 = [[Three-dimensional space|3-dimensional]]
  | list2 = {{Navbox subgroup
+
  | list2 = {{Navbox|subgroup
 
   | group1 = [[centered polyhedral number|centered]]
 
   | group1 = [[centered polyhedral number|centered]]
 
   | list1 =  
 
   | list1 =  
* [[Centered tetrahedral number]]s
+
* [[Centered tetrahedral number|Centered tetrahedral]]
* [[Centered cube number]]s
+
* [[Centered cube number|Centered cube]]
* [[Centered octahedral number]]s
+
* [[Centered octahedral number|Centered octahedral]]
* [[Centered dodecahedral number]]s
+
* [[Centered dodecahedral number|Centered dodecahedral]]
* [[Centered icosahedral number]]s
+
* [[Centered icosahedral number|Centered icosahedral]]
  
 
   | group2 = [[Polyhedral number|non-centered]]
 
   | group2 = [[Polyhedral number|non-centered]]
 
   | list2 =  
 
   | list2 =  
* [[Tetrahedral number]]s
+
* [[Tetrahedral number|Tetrahedral]]
* [[Octahedral number]]s
+
* [[Cube (algebra)|Cubic]]
* [[Dodecahedral number]]s
+
* [[Octahedral number|Octahedral]]
* [[Icosahedral number]]s
+
* [[Dodecahedral number|Dodecahedral]]
* [[Stella octangula number]]s
+
* [[Icosahedral number|Icosahedral]]
 +
* [[Stella octangula number|Stella octangula]]
  
 
   | group3 = [[Pyramidal number|pyramidal]]
 
   | group3 = [[Pyramidal number|pyramidal]]
 
   | list3 =  
 
   | list3 =  
* [[Square pyramidal number]]s
+
* [[Square pyramidal number|Square pyramidal]]
* [[Pentagonal pyramidal number]]s
+
* [[Pentagonal pyramidal number|Pentagonal pyramidal]]
* [[Hexagonal pyramidal number]]s
+
* [[Hexagonal pyramidal number|Hexagonal pyramidal]]
* [[Heptagonal pyramidal number]]s
+
* [[Heptagonal pyramidal number|Heptagonal pyramidal]]
 
}}
 
}}
  
 
   | group3 = [[Four-dimensional space|4-dimensional]]
 
   | group3 = [[Four-dimensional space|4-dimensional]]
   | list3 = {{Navbox subgroup
+
   | list3 = {{Navbox|subgroup
 
   |group1 = centered
 
   |group1 = centered
 
   |list1 =  
 
   |list1 =  
* [[Centered pentachoric number]]s
 
* [[Squared triangular number]]s
 
 
 
   |group2 = non-centered
 
   |group2 = non-centered
 
   |list2 =  
 
   |list2 =  
* [[Pentatope number]]s
+
* [[Pentatope number|Pentatope]]
 +
* [[Squared triangular number|Squared triangular]]
 +
* [[Fourth power|Tesseractic]]
 
}}
 
}}
 
}}
 
}}
 +
 +
|group8 = Combinatorial numbers
 +
|list8 =
 +
* [[Bell number|Bell]]
 +
* [[Cake number|Cake]]
 +
* [[Catalan number|Catalan]]
 +
* [[Dedekind number|Dedekind]]
 +
* [[Delannoy number|Delannoy]]
 +
* [[Euler number|Euler]]
 +
* [[Fuss–Catalan number|Fuss–Catalan]]
 +
* [[Lazy caterer's sequence]]
 +
* [[Lobb number|Lobb]]
 +
* [[Motzkin number|Motzkin]]
 +
* [[Narayana number|Narayana]]
 +
* [[Ordered Bell number|Ordered Bell]]
 +
* [[Schröder number|Schröder]]
 +
* [[Schröder–Hipparchus number|Schröder–Hipparchus]]
 +
 +
|group9 = [[Prime number|Primes]]
 +
|list9 =
 +
* [[Wieferich prime#Wieferich numbers|Wieferich]]
 +
* [[Wall–Sun–Sun prime|Wall–Sun–Sun]]
 +
* [[Wolstenholme prime]]
 +
* [[Wilson prime#Wilson numbers|Wilson]]
  
 
|group10 = [[Pseudoprime]]s
 
|group10 = [[Pseudoprime]]s
 
|list10 =  
 
|list10 =  
* [[Carmichael number]]s
+
* [[Carmichael number]]
* [[Catalan pseudoprime]]s
+
* [[Catalan pseudoprime]]
* [[Elliptic pseudoprime]]s
+
* [[Elliptic pseudoprime]]
* [[Euler pseudoprime]]s
+
* [[Euler pseudoprime]]
* [[Euler–Jacobi pseudoprime]]s
+
* [[Euler–Jacobi pseudoprime]]
* [[Fermat pseudoprime]]s
+
* [[Fermat pseudoprime]]
* [[Fibonacci pseudoprime]]s
+
* [[Frobenius pseudoprime]]
* [[Frobenius pseudoprime]]s
+
* [[Lucas pseudoprime]]
* [[Lucas pseudoprime]]s
+
* [[Lucas–Carmichael number]]
* [[Somer–Lucas pseudoprime]]s
+
* [[Somer–Lucas pseudoprime]]
* [[Strong pseudoprime]]s
+
* [[Strong pseudoprime]]
 +
 
 +
|group11 = [[Arithmetic function]]s and [[Arithmetic dynamics|dynamics]]
 +
|list11 = {{Navbox|subgroup
 +
|group1 = [[Divisor function]]s
 +
|list1 =
 +
* [[Abundant number|Abundant]]
 +
* [[Almost perfect number|Almost perfect]]
 +
* [[Arithmetic number|Arithmetic]]
 +
* [[Betrothed numbers|Betrothed]]
 +
* [[Colossally abundant number|Colossally abundant]]
 +
* [[Deficient number|Deficient]]
 +
* [[Descartes number|Descartes]]
 +
* [[Hemiperfect number|Hemiperfect]]
 +
* [[Highly abundant number|Highly abundant]]
 +
* [[Highly composite number|Highly composite]]
 +
* [[Hyperperfect number|Hyperperfect]]
 +
* [[Multiply perfect number|Multiply perfect]]
 +
* [[Perfect number|Perfect]]
 +
* [[Practical number|Practical]]
 +
* [[Primitive abundant number|Primitive abundant]]
 +
* [[Quasiperfect number|Quasiperfect]]
 +
* [[Refactorable number|Refactorable]]
 +
* [[Semiperfect number|Semiperfect]]
 +
* [[Sublime number|Sublime]]
 +
* [[Superabundant number|Superabundant]]
 +
* [[Superior highly composite number|Superior highly composite]]
 +
* [[Superperfect number|Superperfect]]
  
|group11 = Combinatorial<br/> numbers
+
|group3 = [[Euler's totient function]]
|list11 =
+
|list3 =
* [[Bell number]]s
+
* [[Highly cototient number|Highly cototient]]
* [[Catalan number]]s
+
* [[Highly totient number|Highly totient]]
* [[Dedekind number]]s
+
* [[Noncototient]]
* [[Delannoy number]]s
+
* [[Nontotient]]
* [[Lobb numbers]]
+
* [[Perfect totient number|Perfect totient]]
* [[Motzkin number]]s
+
* [[Sparsely totient number|Sparsely totient]]
* [[Narayana number]]s
 
* [[Ordered Bell number]]s
 
* [[Schröder number]]s
 
* [[Schröder–Hipparchus number]]s
 
  
|group12 = [[Arithmetic function]]s
+
|group2 = [[Prime omega function]]s
|list12 = {{Navbox subgroup
+
|list2 =
  |group1 = By properties of [[Divisor function|σ(''n'')]]
+
* [[Almost prime]]
 +
* [[Semiprime]]
 +
 
 +
|group5 = [[Aliquot sequence]]s
 +
|list5 =
 +
* [[Amicable numbers|Amicable]]
 +
* [[Perfect number|Perfect]]
 +
* [[Sociable numbers|Sociable]]
 +
* [[Untouchable number|Untouchable]]
 +
 
 +
|group6 = [[Primorial]]
 +
|list6 =
 +
* [[Euclid number|Euclid]]
 +
* [[Fortunate number|Fortunate]]
 +
}}
 +
 
 +
|group12 = Other [[prime factor]] or [[divisor]] related numbers
 +
|list12 =
 +
* [[Blum integer|Blum]]
 +
* [[Erdős–Nicolas number|Erdős–Nicolas]]
 +
* [[Erdős–Woods number|Erdős–Woods]]
 +
* [[Friendly number|Friendly]]
 +
* [[Giuga number|Giuga]]
 +
* [[Harmonic divisor number|Harmonic divisor]]
 +
* [[Lucas–Carmichael number|Lucas–Carmichael]]
 +
* [[Pronic number|Pronic]]
 +
* [[Regular number|Regular]]
 +
* [[Rough number|Rough]]
 +
* [[Smooth number|Smooth]]
 +
* [[Sphenic number|Sphenic]]
 +
* [[Størmer number|Størmer]]
 +
* [[Super-Poulet number|Super-Poulet]]
 +
* [[Zeisel number|Zeisel]]
 +
 
 +
|group13 = [[Numeral system]]-dependent numbers
 +
|list13 = {{Navbox|subgroup
 +
  |group1 = [[Arithmetic function]]s and [[Arithmetic dynamics|dynamics]]
 
  |list1 =  
 
  |list1 =  
* [[Abundant number]]s
+
* [[Persistence of a number|Persistence]]
* [[Almost perfect number]]s
+
** [[Additive persistence|Additive]]
* [[Arithmetic number]]s
+
** [[Multiplicative persistence|Multiplicative]]
* [[Colossally abundant number]]s
+
{{Navbox|subgroup
* [[Descartes number]]s
+
  |group1 = [[Digit sum]]
* [[Hemiperfect number]]s
+
  |list1 =
* [[Highly abundant number]]s
+
* [[Digit sum]]
* [[Hyperperfect number]]s
+
* [[Digital root]]
* [[Multiply perfect number]]s
+
* [[Self number|Self]]
* [[Perfect number]]s
+
* [[Sum-product number|Sum-product]]
* [[Primitive abundant number]]s
 
* [[Quasiperfect number]]s
 
* [[Refactorable number]]s
 
* [[Sublime number]]s
 
* [[Superabundant number]]s
 
* [[Superior highly composite number]]s
 
* [[Superperfect number]]s
 
  
|group2 = By properties of [[Arithmetic function#Ω(n), ω(n), νp(n) – prime power decomposition|Ω(''n'')]]
+
  |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]]
 +
* [[Factorion]]
 +
* [[Kaprekar number|Kaprekar]]
 +
* [[Kaprekar's routine|Kaprekar's constant]]
 +
* [[Keith number|Keith]]
 +
* [[Lychrel number|Lychrel]]
 +
* [[Narcissistic number|Narcissistic]]
 +
* [[Perfect digit-to-digit invariant]]
 +
* [[Perfect digital invariant]]
 +
** [[Happy number|Happy]]
 +
}}
 +
|group2 = [[P-adic numbers]]-related
 
  |list2 =
 
  |list2 =
* [[Almost prime]]s
+
* [[Automorphic number|Automorphic]]
* [[Semiprime]]s
+
** [[Trimorphic number|Trimorphic]]
  
  |group3 = By properties of [[Euler's totient function|φ(''n'')]]
+
  |group3 = [[Numerical digit|Digit]]-composition related
 
  |list3 =  
 
  |list3 =  
* [[Highly cototient number]]s
+
* [[Palindromic number|Palindromic]]
* [[Highly totient number]]s
+
* [[Pandigital number|Pandigital]]
* [[Noncototient]]s
+
* [[Repdigit]]
* [[Nontotient]]s
+
* [[Repunit]]
* [[Perfect totient number]]s
+
* [[Self-descriptive number|Self-descriptive]]
* [[Sparsely totient number]]s
+
* [[Smarandache–Wellin number|Smarandache–Wellin]]
 +
* [[Strictly non-palindromic number|Strictly non-palindromic]]
 +
* [[Undulating number|Undulating]]
  
  |group4 = By properties of [[Divisor function#Definition|s(''n'')]]
+
  |group4 = Digit-[[permutation]] related
 
  |list4 =  
 
  |list4 =  
* [[Amicable numbers]]
+
* [[Cyclic number|Cyclic]]
* [[Betrothed numbers]]
+
* [[Digit-reassembly number|Digit-reassembly]]
* [[Deficient number]]s
+
* [[Parasitic number|Parasitic]]
* [[Semiperfect number]]s
+
* [[Primeval number|Primeval]]
 +
* [[Transposable integer|Transposable]]
 +
 
 +
|group5 = Divisor-related
 +
|list5 =
 +
* [[Equidigital number|Equidigital]]
 +
* [[Extravagant number|Extravagant]]
 +
* [[Frugal number|Frugal]]
 +
* [[Harshad number|Harshad]]
 +
* [[Polydivisible number|Polydivisible]]
 +
* [[Smith number|Smith]]
 +
* [[Vampire number|Vampire]]
 +
 
 +
|group6 = Other
 +
|list6 =
 +
* [[Friedman number|Friedman]]
 
}}
 
}}
* [[Euclid number]]s
 
* [[Fortunate number]]s
 
  
|group13 = Dividing a quotient
+
|group14 = [[Binary number]]s
|list13 =  
+
|list14 =
* [[Wieferich number]]s
+
* [[Evil number|Evil]]
* [[Wilson prime#Wilson numbers|Wilson numbers]]
+
* [[Odious number|Odious]]
 +
* [[Pernicious number|Pernicious]]
 +
 
 +
|group15 = Generated via a [[Sieve theory|sieve]]
 +
|list15 =  
 +
* [[Lucky number|Lucky]]
 +
* [[Generation of primes|Prime]]
 +
 
 +
|group16 = [[Sorting algorithm|Sorting]] related
 +
|list16 =
 +
* [[Pancake sorting|Pancake number]]
 +
* [[Sorting number]]
 +
 
 +
|group17 = [[Natural language]] related
 +
|list17 =
 +
* [[Aronson's sequence]]
 +
* [[Ban number|Ban]]
  
|group14 = Other [[prime factor]]<br/> or [[divisor]] related<br/> numbers
+
|group18 = [[Graphemics]] related
|list14 =  
+
|list18 =  
* [[Erdős–Woods number]]s
+
* [[Strobogrammatic number|Strobogrammatic]]
* [[Friendly number]]s
 
* [[Frugal number]]s
 
* [[Giuga number]]s
 
* [[Harmonic divisor number]]s
 
* [[Highly composite number]]s
 
* [[Lucas–Carmichael number]]s
 
* [[Pronic number]]s
 
* [[Regular number]]s
 
* [[Rough number]]s
 
* [[Smooth number]]s
 
* [[Sociable number]]s
 
* [[Sphenic number]]s
 
* [[Størmer number]]s
 
* [[Super-Poulet number]]s
 
  
|group15 = Base-dependent<br/> numbers
+
| belowclass = hlist
|list15 =
+
| belowstyle = font-weight:bold;
* [[Automorphic number]]s
+
| below =
* [[Dudeney number]]s
+
* {{icon|Portal}} [[Portal:Mathematics|Mathematics portal]]
* [[Equidigital number]]s
 
* [[Extravagant number]]s
 
* [[Factorion]]s
 
* [[Friedman number]]s
 
* [[Happy number]]s
 
* [[Harshad number]]s
 
* [[Kaprekar number]]s
 
* [[Keith number]]s
 
* [[Lychrel number]]s
 
* [[Palindromic number]]s
 
* [[Pandigital number]]s
 
* [[Parasitic number]]s
 
* [[Polydivisible number]]s
 
* [[Primeval number]]s
 
* [[Repdigit]]s
 
* [[Repunit]]s
 
* [[Self number]]s
 
* [[Self-descriptive number]]s
 
* [[Smarandache–Wellin number]]s
 
* [[Strictly non-palindromic number]]s
 
* [[Strobogrammatic number]]s
 
* [[Sum-product number]]s
 
* [[Transposable integer]]s
 
* [[Trimorphic number]]s
 
* [[Undulating number]]s
 
* [[Vampire number]]s
 
  
|group16 = [[Recreational mathematics|Recreational<br/> mathematics]]
 
|list16 =
 
* [[Ban number]]s
 
* [[Narcissistic number]]s
 
 
}}<noinclude>
 
}}<noinclude>
 +
{{documentation|content=
 
{{collapsible option}}
 
{{collapsible option}}
  
[[Category:Mathematics templates]]
+
==See also==
 +
*{{tl|Divisor classes}}
 +
*{{tl|Figurate numbers}}
 +
*{{tl|Prime number classes}}
 +
*{{tl|Series (mathematics)}}
 +
 
 +
[[Category:Mathematics navigational boxes]]
 +
}}
 
</noinclude>
 
</noinclude>

Latest revision as of 15:52, 8 September 2020

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See also