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Mathematical formula
single-digit remains, which is called the multiplicative digital root of n {\displaystyle n} . The multiplicative digital root for the first few positive integers
Multiplicative_digital_root
Repeated sum of a number's digits
The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing
Digital_root
Property of a number
is the smallest number of multiplicative persistence 3. In base 10, there is thought to be no number with a multiplicative persistence greater than 11;
Persistence_of_a_number
Figurate number
by a 0 or 5; a final 8 must be preceded by a 2 or 7. In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every
Triangular_number
Sum of a number's digits
decimal digital root of any non-zero integer will be a number in the range 1 to 9, whereas the digit sum can take any value. Digit sums and digital roots
Digit_sum
Number divisible only by 1 and itself
from the multiplicative group of the field to a totally ordered additive group, also called orders), absolute values (certain multiplicative mappings
Prime_number
Type of figurate number
Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine
Hexagonal_number
Number equal to the sum of its proper divisors
until a single digit (called the digital root) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because 8 + 1 +
Perfect_number
Numbers with a certain property involving recursive summation
{\displaystyle b} that eventually reaches 1 when iterated over the perfect digital invariant function for p = 2 {\displaystyle p=2} . The origin of happy
Happy_number
Ten raised to an integer power
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Power_of_10
Numbers obtained by adding the two previous ones
includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However
Fibonacci_sequence
Product of an integer with itself
a square each side of which has the same number of points as the square root of n; thus, square numbers are a type of figurate numbers (other examples
Square_number
Composite number which passes Miller–Rabin primality test
(mod 9). A composite c is called an overpseudoprime base b when the multiplicative order of b mod c × the number of cyclotomic coset (the coset {0} is
Strong_pseudoprime
Power of a prime number
power excluding powers of 2 greater than 4 has a primitive root; thus the multiplicative group of integers modulo pn (that is, the group of units of
Prime_power
Prime number of the form 2^n – 1
congruent to 7 mod 8, so 2 is a quadratic residue mod 2p + 1, and the multiplicative order of 2 mod 2p + 1 must divide ( 2 p + 1 ) − 1 2 = p {\textstyle
Mersenne_prime
Integer having a non-trivial divisor
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Composite_number
Number raised to the third power
perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only 0, 1, and 8. Moreover, the digital root of any number's cube can
Cube_(algebra)
Recursive integer sequence
0 c ( x ) = 1 . {\displaystyle C_{0}=\lim _{x\to 0}c(x)=1\,.} The square root term can be expanded as a power series using the binomial series 1 − 1 −
Catalan_number
Number used for counting
arithmetic operations are defined on natural numbers: addition and multiplication. However, the inverse operations, subtraction and division, only sometimes
Natural_number
Number of form 2^(2^p-1)-1 with prime exponent
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Double_Mersenne_number
Number of orderings allowing ties
2^{n-1}} ordered multiplicative partitions. Numbers that are neither squarefree nor prime powers have a number of ordered multiplicative partitions that
Ordered_Bell_number
Integer filtered out using a sieve similar to that of Eratosthenes
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Lucky_number
Arithmetic operation
invertible elements in a multiplicative monoid, that is, an algebraic structure, with an associative multiplication and a multiplicative identity denoted 1
Exponentiation
Number, non-palindrome after repeated sum with reverse
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Lychrel_number
Number that is less than the sum of its proper divisors
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Abundant_number
Number that is the result of operation on its own digits
numbers are a subset of Friedman numbers where the only operation is a multiplication of two numbers with the same number of digits, for example 1260 = 21
Friedman_number
Square of a triangular number
Row (1893) obtains another proof by summing the numbers in a square multiplication table in two different ways. The sum of the ith row is i times a triangular
Squared_triangular_number
Class of natural numbers with many divisors
Ramanujan (1915). For example, the number with the most divisors per square root of the number itself is 12; this can be demonstrated using some highly composites
Superior highly composite number
Superior_highly_composite_number
Polyhedral number representing a tetrahedron
4.[dubious – discuss] By analogy with the cube root of x, one can define the (real) tetrahedral root of x as the number n such that Ten = x: n = 3 x
Tetrahedral_number
Type of figurate number
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Polygonal_number
Counts pieces of a disk cut by lines
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Lazy_caterer's_sequence
Product of two prime numbers
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Semiprime
Concept in number theory
In number theory, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus
Narcissistic_number
Number used to approximate the square root of 2
comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12
Pell_number
Number that remains the same when its digits are reversed
number whose cube is a palindrome is 2201, and it is a conjecture the fourth root of all the palindrome fourth powers are a palindrome with 100000...000001
Palindromic_number
Odd number with specific properties
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Sierpiński_number
Number that when multiplied by another number moves its last digit to its front
{k}{10n-1}}(10^{m}-1)} where m is the length of the period; i.e. the multiplicative order of 10 modulo (10n − 1). For another example, if n = 2, then 10n
Parasitic_number
Natural number with a decimal representation made of repeated instances of the same digit
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Repdigit
Positive integer of the form (2^(2^n))+1
multiply this by a number A, which is greater than the square root of P and is a primitive root modulo P (i.e., it is not a quadratic residue). Then take
Fermat_number
Type of number introduced by Mike Keith
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Keith_number
Number that cannot be written as an aliquot sum
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Untouchable_number
Count of the possible partitions of a set
numbers, then B n {\displaystyle B_{n}} gives the number of different multiplicative partitions of N {\displaystyle N} . These are factorizations of N {\displaystyle
Bell_number
Integer having only small prime factors
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Smooth_number
Integers occurring in the coefficients of the Taylor series of 1/cosh t
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Euler_number
Size of a geometric arrangement of points
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Figurate_number
Function whose domain is the positive integers
f is multiplicative, then so is g. If f is completely multiplicative, then g is multiplicative, but may or may not be completely multiplicative. There
Arithmetic_function
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
Two raised to an integer power
starting point 2k, and the period is the multiplicative order of 2 modulo 5k, which is φ(5k) = 4 × 5k−1 (see Multiplicative group of integers modulo n).[citation
Power_of_two
Integer divisible by sum of its digits
one as follows: Inserting zeroes into N will not change the sequence of digital sums (just as 21, 201 and 2001 are all 10-Harshad numbers). If we insert
Harshad_number
Odd composite number which passes the given congruence
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Euler_pseudoprime
Number sequence 3,0,2,3,2,5,5,7,10,...
x − 1 = 0 {\displaystyle x^{3}-x-1=0} . If the three solutions are real root α {\displaystyle \alpha } (with approximate value 1.324718 and known
Perrin_number
Numbers that evenly divide powers of 60
roots, such as how the Babylonians found an approximation to the square root of 2, perhaps using regular number approximations of fractions such as 17/12
Regular_number
Result of multiplying four instances of a number together
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Fourth_power
Numbers with many divisors
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Highly_composite_number
Product of the first "n" prime numbers
{\displaystyle \varphi } is the Euler totient function. Any completely multiplicative function is defined by its values at primorials, since it is defined
Primorial
Integer whose multiples are digit rotations
specifically, this sequence is the set of primes p such that b is a primitive root modulo p. A conjecture of Emil Artin is that this sequence contains 37.395
Cyclic_number
Type of Poulet number
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Super-Poulet_number
Number that represents a hexagon with a dot in the center
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Centered_hexagonal_number
Numbers in a type of Lucas sequence
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Jacobsthal_number
Numbers whose prime factors all divide the number more than once
powerful numbers in the interval [1,x]. Then k(x) is proportional to the square root of x. More precisely, c x 1 / 2 − 3 x 1 / 3 ≤ k ( x ) ≤ c x 1 / 2 , c = ζ
Powerful_number
Pair of integers related by their divisors
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Amicable_numbers
Integer whose representation contains every digit in its number base
square numbers. No pandigital cube and no pandigital number with an integer root of higher degree are known. For each natural number n, there exists a maximum
Pandigital_number
Concept in combinatorics
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Cake_number
Type of prime number conjectured to exist
) ≠ k ( p ) {\displaystyle k(p^{2})\neq k(p)} . We have run a test on digital computer which shows that k ( p 2 ) ≠ k ( p ) {\displaystyle k(p^{2})\neq
Wall–Sun–Sun_prime
Centered figurate number
10333, 10837, 11353, and 11881. (sequence A003154 in the OEIS) The digital root of a star number is always 1 or 4, and progresses in the sequence 1,
Star_number
Natural number
generally, in algebra, it denotes the multiplicative identity in any unital ring or field. An element with a multiplicative inverse is called a unit, generalizing
1
Number, product of consecutive integers
constant ℏ = h 2 π {\displaystyle \hbar ={\frac {h}{2\pi }}} times the square root of a half-integer pronic number n 2 {\displaystyle {\frac {n}{2}}} : L =
Pronic_number
Base-dependent property of integers
Inv ( a , c ) {\displaystyle \operatorname {Inv} (a,c)} denote the multiplicative inverse of a {\displaystyle a} modulo c {\displaystyle c} , namely the
Kaprekar_number
Type of natural number
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Colossally_abundant_number
Probable prime that is composite
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Pseudoprime
Area of a right triangle with rational-numbered sides
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Congruent_number
Number of paths between grid corners, allowing diagonal steps
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Delannoy_number
Type of composite number with an even number of digits
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Vampire_number
Two or more natural numbers with a common abundancy index
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Friendly_number
Number of ways to pair up n objects
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Telephone number (mathematics)
Telephone_number_(mathematics)
Numbers that contain only the digit 1
prime, and this n {\displaystyle n} value is r {\displaystyle r} itself or a root of r {\displaystyle r} ) b {\displaystyle b} is not in the form − 4 k 4 {\displaystyle
Repunit
Positive integer that is an integer power of another positive integer
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Perfect_power
Figurate number
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Pentagonal_number
Product of prime numbers, plus one
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Euclid_number
Count of permutations by cycles
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
Numbers k where x - phi(x) = k has many solutions
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Highly_cototient_number
Infinite integer series where the next number is the sum of the two preceding it
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Lucas_number
Number of stacked spheres in a pyramid
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Square_pyramidal_number
Prime such that p^2 divides 2^(p-1)-1
(p − 1)-th degree roots of unity modulo p2 are uniformly distributed in the multiplicative group of integers modulo p2. The following theorem connecting Wieferich
Wieferich_prime
Signal processing phenomenon
speckle noise commonly observed in radar imagery. Examples of multiplicative noise affecting digital photographs are proper shadows due to undulations on the
Multiplicative_noise
Sequence of integers
= 0. {\displaystyle x^{3}-x-1=0.\,} This equation has 3 roots; one real root p (known as the plastic ratio) and two complex conjugate roots q and r. Given
Padovan_sequence
Class of binary number
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Evil_number
Numbers parameterizing ways to partition a set
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Three raised to an integer power
which is a power of two and much smaller. Power of 10 Power of two Square root of 3 Ranucci, Ernest R. (December 1968), "Tantalizing ternary", The Arithmetic
Power_of_three
Polynomial sequence
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Eulerian_number
Numbers whose sum of divisors is twice the number plus 1
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Quasiperfect_number
Concatenation of the first n prime numbers
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Smarandache–Wellin_number
Number that is more than the sum of its proper divisors
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Deficient_number
Combinatorial sequence of numbers
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Dedekind_number
Number whose divisors add to a multiple of that number
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Multiply_perfect_number
Composite number in number theory
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Carmichael_number
Type of composite integer
Persistence Additive Multiplicative Digit sum Digit sum Digital root Self Sum-product Digit product Multiplicative digital root Sum-product Coding-related
Smith_number
Number whose square ends in the same digits
root in roots: for i in range(0, base): new_i = i * base ** (power - 1) + root new_root = polynomial_function(new_i) % pow(base, power) if new_root ==
Automorphic_number
Algorithms for calculating square roots
costly than multiplication, it may be preferable to compute the inverse square root instead. Other methods are available to compute the square root digit by
Square_root_algorithms
Mathematical sequences in combinatorics
so calculating a product entry involves an infinite sum, the matrix multiplications work because these matrices are lower triangular, so only a finite
Stirling_number
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MULTIPLICATIVE DIGITAL-ROOT
MULTIPLICATIVE DIGITAL-ROOT
MULTIPLICATIVE DIGITAL-ROOT
MULTIPLICATIVE DIGITAL-ROOT
MULTIPLICATIVE DIGITAL-ROOT
MULTIPLICATIVE DIGITAL-ROOT
MULTIPLICATIVE DIGITAL-ROOT
MULTIPLICATIVE DIGITAL-ROOT
MULTIPLICATIVE DIGITAL-ROOT
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