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Munchausen number
In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number) is a natural number in a given number base b {\displaystyle
Perfect digit-to-digit invariant
Perfect_digit-to-digit_invariant
Sum of a number's digits
Perfect digital invariant Sideways sum Smith number Sum-product number Bush, L. E. (1940), "An asymptotic formula for the average sum of the digits of
Digit_sum
Concept in number theory
number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number Weisstein, Eric W. "Narcissistic Number". MathWorld. Perfect and PluPerfect
Narcissistic_number
Product of an integer with itself
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with
Square_number
Iterative algorithm on numbers
Meertens number Narcissistic number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number For six-digit numbers, i.e. n=6, (1) 6=3×2
Kaprekar's_routine
Number that is the sum of its own digits, each raised to a given power
theory, a perfect digital invariant (PDI) is a number in a given number base ( b {\displaystyle b} ) that is the sum of its own digits each raised to a given
Perfect_digital_invariant
Number raised to the third power
is a perfect sixth power (in this case 26). The last digits of each 3rd power are: It is, however, easy to show that most numbers are not perfect cubes
Cube_(algebra)
Numbers with a certain property involving recursive summation
over the perfect digital invariant function for p = 2 {\displaystyle p=2} . The origin of happy numbers is not clear. Happy numbers were brought to the attention
Happy_number
Two raised to an integer power
10-choose-3 binary numbers with ten digits that include exactly three 1s). Currently, powers of two are the only known almost perfect numbers. The cardinality of
Power_of_two
Number that is the result of operation on its own digits
numeral system, is the result of a non-trivial expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×
Friedman_number
Repeated sum of a number's digits
(single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum
Digital_root
Number that is the sum of permutations of sub-samples of their own digits
mathematics, the Digit-reassembly numbers, or Osiris numbers, are numbers that are equal to the sum of permutations of sub-samples of their own digits (compare
Digit-reassembly_number
Number that remains the same when its digits are reversed
palindrome) is a number (such as 16361) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical
Palindromic_number
Integer whose multiples are digit rotations
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely-known cyclic
Cyclic_number
Integer divisible by sum of its digits
in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-harshad
Harshad_number
Base-dependent property of integers
constant Meertens number Narcissistic number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number Iannucci (2000) D. R. Kaprekar
Kaprekar_number
Figurate number
equivalent to the handshake problem and fully connected network problems. One way of calculating the depreciation of an asset is the sum-of-years' digits method
Triangular_number
number Kaprekar number Digit sum Persistence of a number Perfect digital invariant Happy number Perfect digit-to-digit invariant Factorion Emirp Palindromic
List of recreational number theory topics
List_of_recreational_number_theory_topics
Integer whose representation contains every digit in its number base
number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 (one billion
Pandigital_number
Sequence in number theory
Kaprekar number Meertens number Narcissistic number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number "Generalized Dudeney Numbers"
Dudeney_number
Positive integer that is an integer power of another positive integer
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as
Perfect_power
Number, non-palindrome after repeated sum with reverse
a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the 196-algorithm
Lychrel_number
Number that represents a hexagon with a dot in the center
rightmost (least significant) digits follow the pattern 1–7–9–7–1 (repeating with period 5). This follows from the last digit of the triangle numbers (sequence
Centered_hexagonal_number
Number whose first n digits is a multiple of n
number base with digits abcde... that has the following properties: Its first digit a is not 0. The number formed by its first two digits ab is a multiple
Polydivisible_number
Prime number of the form 2^n – 1
their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes
Mersenne_prime
Natural number with a decimal representation made of repeated instances of the same digit
instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit". Examples are 11
Repdigit
Number divisible only by 1 and itself
chosen large number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime
Prime_number
Number that when multiplied by another number moves its last digit to its front
results in movement of the last digit of its decimal representation to its front. Here n is itself a single-digit positive natural number. In other
Parasitic_number
Numbers that evenly divide powers of 60
he attributes to Bruins (1970), for generating the six-digit numbers more quickly but that does not generalize in a straightforward way to larger values
Regular_number
Numbers that contain only the digit 1
mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for "repeated unit"
Repunit
Mathematical formula
is found by multiplying the digits of n {\displaystyle n} together, then repeating this operation until only a single-digit remains, which is called the
Multiplicative_digital_root
Numbers with many divisors
composite number that is not a Harshad number is 245,044,800; it has a digit sum of 27, which does not divide evenly into 245,044,800. 10 of the first
Highly_composite_number
Number that is the sum of the factorials of its digits
Kaprekar number Meertens number Narcissistic number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number Sloane, Neil, "A014080", On-Line
Factorion
Type of number introduced by Mike Keith
Fibonacci-like digit) is a natural number n {\displaystyle n} in a given number base b {\displaystyle b} with k {\displaystyle k} digits such that when
Keith_number
Recursive integer sequence
digits greater than p + 1/2; also count digits equal to p + 1/2 unless final; and count digits equal to p − 1/2 if not final and the next digit
Catalan_number
Numeral ambigram
2002 is not a strobogrammatic number due to the 2 being different in traditional fonts. Using only the digits 0, 1, 6, 8 and 9, the next upside-down year
Strobogrammatic_number
Number equal to the sum of its proper divisors
itself has to be prime. He also says (wrongly) that the perfect numbers end in 6 or 8 alternately. (The first 5 perfect numbers end with digits 6, 8, 6,
Perfect_number
Number whose divisors add to a multiple of that number
called k-perfect (or k-fold perfect) if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and
Multiply_perfect_number
Integer describing itself
integer m in a given base b that is b digits long, and each digit d at position n (the most significant digit being at position 0 and the least significant
Self-descriptive_number
Class of natural numbers with many divisors
ratio between the number of divisors an integer has and that integer raised to some positive power. For any possible exponent, whichever integer has the
Superior highly composite number
Superior_highly_composite_number
Ten raised to an integer power
Numbers larger than about a trillion are rarely referred to by name or written out as digits, but instead are typically described with exponent notation
Power_of_10
Number that is its own Gödel number
constant Kaprekar number Narcissistic number Perfect digit-to-digit invariant Perfect digital invariant Sum-product number Richard S. Bird (1998). "Meertens
Meertens_number
Three raised to an integer power
make an ideal system of coins. In number theory, all powers of three are perfect totient numbers. The sums of distinct powers of three form a Stanley sequence
Power_of_three
Number whose square ends in the same digits
(sometimes referred to as a circular number) is a natural number in a given number base b {\displaystyle b} whose square "ends" in the same digits as the number
Automorphic_number
Number of dots in a centred dot square
centered square numbers are odd, and in base 10 one can notice the one's digit follows the pattern 1-5-3-5-1. All centered square numbers and their divisors
Centered_square_number
Positive integer of the form (2^(2^n))+1
where p is an odd prime. With the exception of F0 and F1, the last decimal digit of a Fermat number is 7. The sum of the reciprocals of all the Fermat numbers
Fermat_number
Result of multiplying five instances of a number together
in the OEIS) For any integer n, the last decimal digit of n5 is the same as the last (decimal) digit of n, i.e. n ≡ n 5 ( mod 10 ) {\displaystyle n\equiv
Fifth_power_(algebra)
Product of prime numbers, plus one
En ≥ 3 + 1 have a final digit of 1. Likewise, the last digit of every Kummer number is 9. No Euclid or Kummer numbers are perfect powers. Unsolved problem
Euclid_number
Number equal to the product of the sum and product of its digits
number Perfect digit-to-digit invariant Perfect digital invariant Sloane, N. J. A. (ed.). "Sequence A038369 (Numbers n such that n = (product of digits of
Sum-product_number
Result of multiplying four instances of a number together
A000583 in the OEIS). The last digit of a fourth power in decimal can only be 0, 1, 5, or 6. In hexadecimal the last nonzero digit of a fourth power is always
Fourth_power
Type of composite integer
number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case
Smith_number
Triangular array of natural numbers
{\displaystyle \operatorname {N} (n,1)=1} , since the only way to get a single sub-pattern () is to have all the opening parentheses in the first n {\displaystyle
Narayana_number
Number that is less than the sum of its proper divisors
{n}{20}}=n+{\tfrac {n}{10}}.} Similarly, every multiple of a perfect number (except the perfect number itself) is abundant. For example, every multiple n
Abundant_number
Number that is more than the sum of its proper divisors
Closely related to deficient numbers are perfect numbers with σ(n) = 2n, and abundant numbers with σ(n) > 2n. Nicomachus was the first to subdivide numbers
Deficient_number
Integer having a non-trivial divisor
factors of a number are repeated it is called a powerful number (too, all perfect powers are powerful numbers). If none of its prime factors are repeated
Composite_number
Product of two prime numbers
a star cluster. It consisted of 1679 {\displaystyle 1679} binary digits intended to be interpreted as a 23 × 73 {\displaystyle 23\times 73} bitmap image
Semiprime
Numbers obtained by adding the two previous ones
Fibonacci numbers with d decimal digits. More generally, in the base b representation, the number of digits in Fn is asymptotic to n log b φ = n log φ log
Fibonacci_sequence
Type of figurate number
pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula M p 2 p − 1 = M p M p + 1 2 =
Hexagonal_number
Probable prime that is composite
in 1988 that it would cost $10 million to factor a number with 144 digits, and $100 billion to factor a 200-digit number (the cost today is dramatically
Pseudoprime
Integer filtered out using a sieve similar to that of Eratosthenes
number in a set which is generated by a certain "sieve". This sieve is similar to the sieve of Eratosthenes that generates the primes, but it eliminates numbers
Lucky_number
Type of Poulet number
In number theory, a super-Poulet number is a Poulet number, or pseudoprime to base 2, whose every divisor d {\displaystyle d} divides 2 d − 2 {\displaystyle
Super-Poulet_number
Centered figurate number
is always 1 or 4, and progresses in the sequence 1, 4, 1. The last two digits of a star number in base 10 are always 01, 13, 21, 33, 37, 41, 53, 61, 73
Star_number
Type of natural number
sum of any other natural number n {\displaystyle n} and the individual digits of n {\displaystyle n} . 20 is a self number (in base 10), because no such
Self_number
Number that permute or shift cyclically when multiplied by another number
or directly. For any integer coprime to 10, its reciprocal is a repeating decimal without any non-recurring digits. E.g. 1⁄143 = 0.006993006993006993.
Transposable_integer
Square of a triangular number
JSTOR 2688231. Stroeker, R. J. (1995), "On the sum of consecutive cubes being a perfect square", Compositio Mathematica, 97 (1–2): 295–307, MR 1355130. Toeplitz
Squared_triangular_number
Numbers k where x - phi(x) = k has many solutions
positive integer k {\displaystyle k} which is above 1 and has more solutions to the equation x − ϕ ( x ) = k {\displaystyle x-\phi (x)=k} than any other integer
Highly_cototient_number
Property of a number
has to replace the number by the sum or product of its digits until one reaches a single digit. Because the numbers are broken down into their digits, the
Persistence_of_a_number
Composite number in number theory
fact, composite. Arnault gives a 397-digit Carmichael number N {\displaystyle N} that is a strong pseudoprime to all prime bases less than 307: N = p
Carmichael_number
Integer having only small prime factors
n-friable) number is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number is a number in which every prime factor
Smooth_number
Arithmetic operation
base ten (decimal) number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and
Exponentiation
Infinite integer series where the next number is the sum of the two preceding it
has 30950 decimal digits. As of August 2022[update], the largest known Lucas probable prime is L5466311, with 1,142,392 decimal digits. If Ln is prime then
Lucas_number
Odd number with specific properties
2931991, ... (sequence A076336 in the OEIS). The number 78557 was proved to be a Sierpiński number by John Selfridge in 1962, who showed that all numbers
Sierpiński_number
Type of composite number with an even number of digits
natural number with an even number of digits, that can be factored into two natural numbers each with half as many digits as the original number, where the
Vampire_number
Number that is the sum of its iterated totients
theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, one applies the totient function to a number
Perfect_totient_number
Abundant number whose proper divisors are all deficient numbers
having no abundant proper divisor, which can also include divisors that are perfect numbers. It starts: 12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114
Primitive_abundant_number
Type of figurate number
sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between
Polygonal_number
Concatenation of the first n prime numbers
three are 2, 23 and 2357 (sequence A069151 in the OEIS). The fourth is 355 digits long: it is the result of concatenating the first 128 prime numbers, through
Smarandache–Wellin_number
Number whose sums of distinct divisors represent all smaller numbers
characterization makes it possible to determine whether a number is practical by examining its prime factorization. Every even perfect number and every power of
Practical_number
Numbers whose sum of divisors is twice the number minus 1
Do any non-power of 2 almost perfect numbers exist? More unsolved problems in mathematics In mathematics, an almost perfect number (sometimes also called
Almost_perfect_number
Integer of the form 3 × 2^n – 1 for non-negative n
the first to study these numbers and their relation to amicable numbers. The binary representation of the Thabit number 3·2n−1 is n+2 digits long, consisting
Thabit_number
Class of natural numbers
first exception is the 105th superabundant number, 149602080797769600. The digit sum is 81, but 81 does not divide evenly into this superabundant number
Superabundant_number
Polyhedral number representing a tetrahedron
numbers are also perfect squares, namely: Te1 = 12 = 1 Te2 = 22 = 4 Te48 = 1402 = 19600. Like square numbers are congruent to 0, 1, or 4 (mod
Tetrahedral_number
Number of form 2^(2^p-1)-1 with prime exponent
"Martian prime". Cunningham chain Double exponential function Fermat number Perfect number Wieferich prime Chris Caldwell, Mersenne Primes: History, Theorems
Double_Mersenne_number
American mathematician
Reacher novel series in the book Never Go Back, which uses perfect digit-to-digit invariant numbers in the plot: "Such numbers had been much discussed
Joseph_Madachy
Natural number
nn: 3413 = 11 + 22 + 33 + 44 + 55 3435 – a perfect digit-to-digit invariant, equal to the sum of its digits to their own powers (33 + 44 + 33 + 55 = 3435)
3000_(number)
Type of natural number
number itself; 1+2+3=6, so this condition is not met (and 6 is instead a perfect number). However all colossally abundant numbers are also superabundant
Colossally_abundant_number
Size of a geometric arrangement of points
formula for all triangular numbers that are also perfect squares, among many other discoveries relating to figurate numbers. Figurate numbers have played
Figurate_number
Integer that is both a perfect square and a triangular number
number, in other words, the sum of all integers from 1 {\displaystyle 1} to n {\displaystyle n} has a square root that is an integer. There are infinitely
Square_triangular_number
Class of binary number
numbers from 0 {\displaystyle 0} to 2 k − 1 {\displaystyle 2^{k}-1} , for any k {\displaystyle k} , provides a solution to the Prouhet–Tarry–Escott problem
Evil_number
Number used for counting
week", in which case they are called cardinal numbers. They are also used to label places in an ordered series, such as: "the third day of the month",
Natural_number
Figurate number
to just check if 24x + 1 is a perfect square. For non-generalized pentagonal numbers, in addition to the perfect square test, it is also required to check
Pentagonal_number
Number that cannot be written as an aliquot sum
composite numbers (since except 2, all even numbers are composite). No perfect number is untouchable, since, at the very least, it can be expressed as
Untouchable_number
Type of natural number in recreational number theory
with n digits is 1, 4, 11, 31, 106, 402, 1953, 10542, 64905, 362451, 2970505, ... (sequence A076730 in the OEIS) The smallest n-digit number to achieve
Primeval_number
Centered figurate number that represents a nonagon with a dot in the center
every Mersenne prime greater than 3 is congruent to 1 modulo 3, it follows that every even perfect number greater than 6 is a centered nonagonal number
Centered_nonagonal_number
Number that has a perfect number of factors adding up to another perfect number
integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number. The number 12
Sublime_number
Positive integer that is the product of three distinct prime numbers
process at a minimum adding another prime factor, or raising an existing factor to a higher power. The first case of two consecutive sphenic integers is 230
Sphenic_number
Integer sequence in number theory
theory, a Descartes number is an odd number which would have been an odd perfect number if one of its composite factors were prime. They are named after
Descartes_number
Integers occurring in the coefficients of the Taylor series of 1/cosh t
order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive (sequence A000364 in the OEIS). This article adheres to the
Euler_numbers
Two or more natural numbers with a common abundancy index
Numbers with abundancy 2 are also known as perfect numbers. There are several unsolved problems related to the friendly numbers. In spite of the similarity
Friendly_number
Number of unique ways to draw non-intersecting chords in a circle
figure shows the 9 ways to draw non-intersecting chords between 4 points on a circle (M4 = 9): The following figure shows the 21 ways to draw non-intersecting
Motzkin_number
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