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Number that permute or shift cyclically when multiplied by another number
In mathematics, the transposable integers are integers that permute or shift cyclically when they are multiplied by another integer n {\displaystyle n}
Transposable_integer
Operation in music
another pitch. The transposition of a set A by n semitones is designated by Tn(A), representing the addition (modulo 12) of an integer n to each of the
Transposition_(music)
Number used for counting
are 0 (if included), 1, 2, 3, and so on. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
Mathematical optimization problem restricted to integers
An integer programming, also known as integer optimization, problem is a mathematical optimization or feasibility program in which some or all of the variables
Integer_programming
Number that when multiplied by another number moves its last digit to its front
A128858 in the OEIS) Cyclic number Linear-feedback shift register Transposable integer Dawidoff, Nicholas (March 25, 2009), "The Civil Heretic", New York
Parasitic_number
Property of being an even or odd number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not. For
Parity_(mathematics)
Integer side lengths of a right triangle
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known
Pythagorean_triple
Mathematical logic concept
positive integer N is a non-square number, its square root is irrational, we can equivalently prove its contrapositive, that if a positive integer N has
Contraposition
Problem in computer science
non-singleton cycles in the in-situ transposition of a rectangular j X k matrix)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane
In-place_matrix_transposition
Sporadic simple group
Sloane, N. J. A. (ed.). "Sequence A007267". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Gorenstein, D. (1993), "A brief history of
Baby_monster_group
Type of (mathematical) permutation with no fixed element
(product) of transpositions—formally, they are generators for the group. In fact, when the set being permuted is {1, 2, ..., n} for some integer n, then any
Cyclic_permutation
Integer having a non-trivial divisor
number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly, it is a positive integer that has at least one
Composite_number
Metric in computer science
length(a)], b[1..length(b)] output: distance, integer let d[0..length(a), 0..length(b)] be a 2-d array of integers, dimensions length(a)+1, length(b)+1 // note
Damerau–Levenshtein_distance
Product of two distinct primes ≡ 3 (mod 4)
form 4t + 3, for some integer t. Integers of this form are referred to as Blum primes. This means that the factors of a Blum integer are Gaussian primes
Blum_integer
Rational numbers with root 5 added
} , integer primes of the form p = 5 n ± 2 {\displaystyle p=5n\pm 2} where n {\displaystyle n} is an integer, and the factors of integer primes
Golden_field
Product of an integer with itself
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 itself. For example, 9 is
Square_number
Numbers with many divisors
a positive integer that has more divisors than all smaller positive integers. If d(n) denotes the number of divisors of a positive integer n, then a positive
Highly_composite_number
Number that is less than the sum of its proper divisors
excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number:
Abundant_number
Two raised to an integer power
of the form 2n where n is an integer, that is, the result of exponentiation with the number two as the base and integer n as the exponent. In the fast-growing
Power_of_two
Matrix form in linear algebra
normal form is an analogue of reduced echelon form for matrices over the integers Z {\displaystyle \mathbb {Z} } . Just as reduced echelon form can be used
Hermite_normal_form
Ten raised to an integer power
of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition
Power_of_10
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix
Unimodular_matrix
Topics referred to by the same term
set is even if it is composed of an even number of transpositions Singly even number, an integer divisible by 2 but not divisible by 4 Even code, if
Even
Natural sorting algorithm
Python's bools can be # evaluated as integers; True == 1 and False == 0. return_list.append(sum(n > i for n in transposed_list)) # The resulting list is sorted
Bead_sort
positive integer, n! is the product of the first n positive integers, and is read as "n factorial". 2. Double factorial: if n is a positive integer, n!!
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Three raised to an integer power
number of the form 3n where n is an integer, that is, the result of exponentiation with number three as the base and integer n as the exponent. The first ten
Power_of_three
Number that cannot be written as an aliquot sum
untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are
Untouchable_number
Sum of a number's digits
sum of the base 10 digits of the integers 0, 1, 2, ... is given by OEIS: A007953 in the On-Line Encyclopedia of Integer Sequences. Borwein & Borwein (1992)
Digit_sum
Integer having only small prime factors
In number theory, an n-smooth (or n-friable) number is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number
Smooth_number
Positive integer that is an integer power of another positive integer
factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally
Perfect_power
Number divisible only by 1 and itself
trial division, tests whether n {\displaystyle n} is a multiple of any integer between 2 and n {\displaystyle {\sqrt {n}}} . Faster algorithms include
Prime_number
Matrix whose conjugate transpose is its negative (additive inverse)
Hermitian if k {\displaystyle k} is an even integer and skew-Hermitian if k {\displaystyle k} is an odd integer. A {\displaystyle A} is skew-Hermitian if
Skew-Hermitian_matrix
Alphanumeric code that uniquely identifies a bank account in any participating country
unsigned integers can accommodate all valid IBAN values. 231 is approximately equal to 2.1 × 109, making it possible for any 9-digit integer to be handled
International Bank Account Number
International_Bank_Account_Number
Class of natural numbers with many divisors
number of divisors an integer has and that integer raised to some positive power. For any possible exponent, whichever integer has the greatest ratio
Superior highly composite number
Superior_highly_composite_number
Positive integer whose divisors have a harmonic mean that is an integer
divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are
Harmonic_divisor_number
Arithmetic operation
involving two numbers: the base, b, and the exponent, n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that
Exponentiation
Prime number of the form 2^n – 1
of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied
Mersenne_prime
Infinite integer series where the next number is the sum of the two preceding it
The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and
Lucas_number
Integer where the average of its positive divisors is also an integer
number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number
Arithmetic_number
Way of arriving to a mathematical proof
∎ By definition, if n is an odd integer, it can be expressed as n = 2 k + 1 {\displaystyle n=2k+1} for some integer k. Thus n 2 = ( 2 k + 1 ) 2 = ( 2
Direct_proof
Class of binary number
In number theory, an evil number is a non-negative integer that has an even number of 1s in its binary expansion. These numbers give the positions of
Evil_number
Topics referred to by the same term
Odd may refer to: Even and odd numbers, an integer is odd if dividing by two does not yield an integer Even and odd functions, a function is odd if
Odd
Square root of the determinant of a skew-symmetric square matrix
as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depends on m. When m is odd, the polynomial is zero
Pfaffian
Integer filtered out using a sieve similar to that of Eratosthenes
(Numbers that are both lucky and prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Guy, Richard K. (2004). Unsolved problems in
Lucky_number
Type of group in abstract algebra
irreducible representation can be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed
Symmetric_group
Number raised to the third power
cube of an integer. The non-negative perfect cubes up to 603 are (sequence A000578 in the OEIS): Geometrically speaking, a positive integer m is a perfect
Cube_(algebra)
Number equal to the sum of its proper divisors
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number
Perfect_number
Solving an optimization problem with a quadratic objective function
elements of the vector x will need to take on integer values. This leads to the formulation of a mixed-integer quadratic programming (MIQP) problem. Applications
Quadratic_programming
Type of positive integer
theory, a positive integer k is said to be an Erdős–Woods number if it has the following property: there exists a positive integer a such that in the
Erdős–Woods_number
Instruction set extension by Intel
16-bit integer operations IFMA, VBMI: introduced with Cannon Lake. AVX-512 Integer Fused Multiply Add (IFMA) – fused multiply add of integers using 52-bit
AVX-512
Concept in number theory
numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer. Arithmetic dynamics Dudeney number Factorion
Narcissistic_number
Positive integer that is the product of three distinct prime numbers
theory, a sphenic number (from Ancient Greek: σφήν, 'wedge') is a positive integer that is the product of three distinct prime numbers. For example, since
Sphenic_number
Number with a half-integer abundancy index
hemiperfect number is a positive integer with a half-integer abundancy index. In other words, σ(n)/n = k/2 for an odd integer k, where σ(n) is the sum-of-divisors
Hemiperfect_number
Combinatorial object in representation theory
the number of boxes in each row gives a partition λ of a non-negative integer n, the total number of boxes of the diagram. The Young diagram is said
Young_tableau
Geometric arrangements of points, foundational to Lie theory
\beta } onto the line through α {\displaystyle \alpha } is an integer or half-integer multiple of α {\displaystyle \alpha } . Equivalent ways of writing
Root_system
Mathematical concept
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. When k
Lucky_numbers_of_Euler
Mapping arbitrary data to fixed-size values
32-bit integer. Thus the 32-bit integer Integer and 32-bit floating-point Float objects can simply use the value directly, whereas the 64-bit integer Long
Hash_function
Proprietary array programming language
symbol `john q)50 / an atom of type integer 50 q)`john`jack / a list of symbols `john`jack q)50 60 / a list of integers 50 60 q)`john`jack!50 60 / a list
Q (programming language from Kx Systems)
Q_(programming_language_from_Kx_Systems)
Numbers k where x - phi(x) = k has many solutions
theory, a branch of mathematics, a highly cototient number is a positive integer k {\displaystyle k} which is above 1 and has more solutions to the equation
Highly_cototient_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 is the
Cyclic_number
Result of multiplying four instances of a number together
fourth power is always 1. Every positive integer can be expressed as the sum of at most 19 fourth powers; every integer larger than 13792 can be expressed as
Fourth_power
Integer of the form 3 × 2^n – 1 for non-negative n
Qurra number, or 321 number is an integer of the form 3 ⋅ 2 n − 1 {\displaystyle 3\cdot 2^{n}-1} for a non-negative integer n. The first few Thabit numbers
Thabit_number
Mathematical problem
Since the determinant of a {0, 1} matrix is an integer, the determinant of an n×n {1, −1} matrix is an integer multiple of 2n−1. Let R be an n by n {1, −1}
Hadamard's maximal determinant problem
Hadamard's_maximal_determinant_problem
Classification of pitch class sets
may be denoted by an integer in the range from 0 to 11 (inclusive), and a pitch class set may be denoted by a set of these integers. The prime form of a
Forte_number
Number of paths between grid corners, allowing diagonal steps
{\displaystyle m} and n {\displaystyle n} , the points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, and
Delannoy_number
Numbers whose prime factors all divide the number more than once
is the product of a square and a cube. A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently
Powerful_number
Positive integer with large prime factors
A k-rough number, as defined by Finch in 2001 and 2003, is a positive integer whose prime factors are all greater than or equal to k. k-roughness has
Rough_number
Product of prime numbers, plus one
In mathematics, Euclid numbers are integers of the form En = pn # + 1, where pn # is the nth primorial (the product of the first n prime numbers). They
Euclid_number
Integer divisible by the number of its divisors
A refactorable number or tau number is an integer n that is divisible by the count of its divisors, or to put it algebraically, n is such that τ ( n )
Refactorable_number
Numbers with special prime factorization
number is a number that is powerful but not a perfect power. A positive integer n is a powerful number if, for every prime factor p of n, p2 is also a
Achilles_number
Unique numeric book identifier since 1970
the SBN without the zero. The check digit is base eleven, and can be an integer between 0 and 9, or an 'X'. The system for 13-digit ISBNs is not compatible
ISBN
Number equal to the product of the sum and product of its digits
numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer. The example below implements the sum-product
Sum-product_number
Type of positive composite integer
In mathematics, a Lucas–Carmichael number is a positive composite integer n such that If p is a prime factor of n, then p + 1 is a factor of n + 1; n
Lucas–Carmichael_number
Reference point for tuning musical instruments
integer for simplicity and convenience. In principle, this allows for playing along with modern fixed-pitch instruments if their parts are transposed
Concert_pitch
Concatenation of the first n prime numbers
In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base
Smarandache–Wellin_number
Type of positive integer pairs
number theory, betrothed numbers or quasi-amicable numbers are two positive integers such that the sum of the proper divisors of either number is one more than
Betrothed_numbers
Abstract algebra concept
{\displaystyle q} are integers with gcd(p, q) = 1, then { p , q } {\displaystyle \{p,q\}} also generates the group of integers under addition by Bézout's
Generating_set_of_a_group
Figurate number
The triangular numbers or triangle numbers are the sequence of positive integers that can be represented as a lattice of points arranged in an equilateral
Triangular_number
Mathematical concept in prime numbers
called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 (where x2 is relatively
Idoneal_number
Positive integer of the form (2^(2^n))+1
them, is a positive integer of the form: F n = 2 2 n + 1 , {\displaystyle F_{n}=2^{2^{n}}+1,} where n is a non-negative integer. The first few Fermat
Fermat_number
Positive integer of the form 4n + 1
number theory, a branch of mathematics, a Hilbert number is a positive integer of the form 4n + 1 (Flannery & Flannery (2000, p. 35)). The Hilbert numbers
Hilbert_number
Type of natural number
between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio
Colossally_abundant_number
Mathematical structure in abstract algebra
trivially-*-ring. The * flips the sign of that square root. A quadratic integer ring (for some D) is a commutative *-ring with the * defined in the similar
*-algebra
Number, product of consecutive integers
A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n ( n + 1 ) {\displaystyle n(n+1)} . The study
Pronic_number
Integer that is both a perfect square and a triangular number
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 many square
Square_triangular_number
Base-dependent property of integers
K ( N ) {\displaystyle K(N)} for a given integer N {\displaystyle N} can be defined as the set of integers X {\displaystyle X} for which there exist
Kaprekar_number
Number sequence 3,0,2,3,2,5,5,7,10,...
mathematics, the Perrin numbers are a doubly infinite constant-recursive integer sequence with characteristic equation x3 = x + 1. The Perrin numbers, named
Perrin_number
Pair of integers related by their divisors
\\q&=3\times 2^{n}-1,\\r&=9\times 2^{2n-1}-1,\end{aligned}}} where n > 1 is an integer and p, q, r are prime numbers, then 2n × p × q and 2n × r are a pair of
Amicable_numbers
Mathematical operation with only one operand
− ( − 3 ) = 3 {\displaystyle -(-3)=3} For any positive integer n, the product of the integers less than or equal to n is a unary operation called factorial
Unary_operation
Set of rules defining correctly structured programs
interpreted according to use. For example, ⌊3.2 gives 3, the largest integer not above the argument, and 3⌊2 gives 2, the lower of the two arguments
APL_syntax_and_symbols
Number of the form x^2 + xy + y^2
In number theory, the numbers of the form x2 + xy + y2 for integer x, y are called the Löschian numbers (or Loeschian numbers). These numbers are named
Löschian_number
Numbers in a type of Lucas sequence
In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers
Jacobsthal_number
Graph with oriented edges
arc of (x, y). The adjacency matrix of a multidigraph with loops is the integer-valued matrix with rows and columns corresponding to the vertices, where
Directed_graph
Matrix equal to its conjugate-transpose
matrix with complex-valued entries that is equal to its own conjugate transpose. That is, if the element in the j {\displaystyle j} -th row and k
Hermitian_matrix
Numbers whose sum of divisors is twice the number minus 1
even almost perfect numbers are those of the form 2k for some positive integer k; however, it has not been shown that all almost perfect numbers are of
Almost_perfect_number
Python library for numerical programming
subroutine ftest(a, b, n, c, d) implicit none integer, intent(in) :: a, b, n integer, intent(out) :: c, d integer :: i c = 0 do i = 1, n c = a + b + c end
NumPy
Mathematical formula
{\displaystyle n} . The multiplicative digital root for the first few positive integers are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2,
Multiplicative_digital_root
Associative array for storing key–value pairs
function in which an unsigned integer that is initially zero is repeatedly left shifted one bit and then xor'ed with the integer value of the next character
Hash_table
Unsolved problem in mathematics
{\displaystyle \mathbb {Q} } is the cyclic group Z/nZ for any positive integer n. To do this, choose a prime p such that p ≡ 1 (mod n); this is possible
Inverse_Galois_problem
Number whose sums of distinct divisors represent all smaller numbers
number or panarithmic number is a positive integer n {\displaystyle n} such that all smaller positive integers can be represented as sums of distinct divisors
Practical_number
travel, tourism, insurance
TRANSPOSABLE INTEGER
TRANSPOSABLE INTEGER
TRANSPOSABLE INTEGER
TRANSPOSABLE INTEGER
TRANSPOSABLE INTEGER
TRANSPOSABLE INTEGER
TRANSPOSABLE INTEGER
TRANSPOSABLE INTEGER
TRANSPOSABLE INTEGER
travel, tourism, insurance