In recreational math, a repdigit or a monodigit[1] is a type of natural number. It is only made of the same repeated digit. The word is a portmanteau of repeated and digit. Examples can be numbers like 11, 666, 4444, and 999999. Repdigits are palindromic numbers (read the same forwards and backwards) and are multiples of repunits (a number that only has 1 in it).
Repdigits are the written in base of the number where x is the digit that is repeated (), and is how many times that number repeats. For example, the repdigit 77777 in base 10 is .
Brazilian numbers are another way of making repdigits. These are numbers that can be written as a repdigit in any base. A Brazilian Number can't be the repdigit 11 and it can't be a number with only one digit. 27 would be a Brazilian number because 27 is the repdigit 33 in base 8. 9 is not a Brazilian number because its only repdigit is 118.[2] The first twenty Brazilian numbers are
- 7, 8, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 33, ... (sequence A125134 in the OEIS).
Repdigits were studied since 1974.[3] In Beiler (1966), it was originally called "monodigit numbers".[1] Brazilian numbers were created later in 1994 in the 9th Iberoamerican Mathematical Olympiad. It took place in Fortaleza, Brazil. The first problem in this competition, proposed by Mexico, was this:[4]
"A number n > 0 is called "Brazilian" if there exists an integer b such that 1 < b < n – 1 for which the representation of n in base b is written with all equal digits. Prove that 1994 is Brazilian and that 1993 is not Brazilian."
|
---|
|
|
Other polynomial numbers |
---|
- Hilbert
- Idoneal
- Leyland
- Loeschian
- Lucky numbers of Euler
|
|
|
---|
- Fibonacci
- Jacobsthal
- Leonardo
- Lucas
- Padovan
- Pell
- Perrin
|
|
Possessing a specific set of other numbers |
---|
- Congruent
- Knödel
- Riesel
- Sierpiński
|
|
Expressible via specific sums |
---|
- Nonhypotenuse
- Polite
- Practical
- Primary pseudoperfect
- Ulam
- Wolstenholme
|
|
Figurate numbers |
---|
2-dimensional | centered |
- Centered triangular
- Centered square
- Centered pentagonal
- Centered hexagonal
- Centered heptagonal
- Centered octagonal
- Centered nonagonal
- Centered decagonal
- Star
|
---|
non-centered |
- Triangular
- Square
- Square triangular
- Pentagonal
- Hexagonal
- Heptagonal
- Octagonal
- Nonagonal
- Decagonal
- Dodecagonal
|
---|
|
---|
3-dimensional | centered |
- Centered tetrahedral
- Centered cube
- Centered octahedral
- Centered dodecahedral
- Centered icosahedral
|
---|
non-centered |
- Tetrahedral
- Cubic
- Octahedral
- Dodecahedral
- Icosahedral
- Stella octangula
|
---|
pyramidal | |
---|
|
---|
4-dimensional | non-centered |
- Pentatope
- Squared triangular
- Tesseractic
|
---|
|
---|
|
|
Combinatorial numbers |
---|
- Bell
- Cake
- Catalan
- Dedekind
- Delannoy
- Euler
- Eulerian
- Fuss–Catalan
- Lah
- Lazy caterer's sequence
- Lobb
- Motzkin
- Narayana
- Ordered Bell
- Schröder
- Schröder–Hipparchus
- Stirling first
- Stirling second
|
|
|
---|
- Wieferich
- Wall–Sun–Sun
- Wolstenholme prime
- Wilson
|
|
Pseudoprimes |
---|
- 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
|
|
Arithmetic functions and dynamics |
---|
Divisor functions | |
---|
Prime omega functions | |
---|
Euler's totient function |
- Highly cototient
- Highly totient
- Noncototient
- Nontotient
- Perfect totient
- Sparsely totient
|
---|
Aliquot sequences |
- Amicable
- Perfect
- Sociable
- Untouchable
|
---|
Primorial | |
---|
|
|
|
---|
- Blum
- Cyclic
- Erdős–Nicolas
- Erdős–Woods
- Friendly
- Giuga
- Harmonic divisor
- Lucas–Carmichael
- Pronic
- Regular
- Rough
- Smooth
- Sphenic
- Størmer
- Super-Poulet
- Zeisel
|
|
|
---|
Arithmetic functions and dynamics | Digit sum |
- Digit sum
- Digital root
- Self
- Sum-product
|
---|
Digit product |
- Multiplicative digital root
- Sum-product
|
---|
Coding-related | |
---|
Other |
- Dudeney
- Factorion
- Kaprekar
- Kaprekar's constant
- Keith
- Lychrel
- Narcissistic
- Perfect digit-to-digit invariant
- Perfect digital invariant
|
---|
|
---|
P-adic numbers-related | |
---|
Digit-composition related |
- Palindromic
- Pandigital
- Repdigit
- Repunit
- Self-descriptive
- Smarandache–Wellin
- Strictly non-palindromic
- Undulating
|
---|
Digit-permutation related |
- Cyclic
- Digit-reassembly
- Parasitic
- Primeval
- Transposable
|
---|
Divisor-related |
- Equidigital
- Extravagant
- Frugal
- Harshad
- Polydivisible
- Smith
- Vampire
|
---|
Other | |
---|
|
|
|
|
|
---|
- Pancake number
- Sorting number
|
|
|
|