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Number without repeated prime factors
In mathematics, a square-free integer (or squarefree integer) is an integer that is divisible by no square number other than 1. That is, its prime factorization
Square-free_integer
Root of a quadratic polynomial with a unit leading coefficient
by the square root of the unique square-free integer D that satisfies b2 − 4c = De2 for some integer e. If D is positive, the quadratic integer is real
Quadratic_integer
Algebraic construction
is given by (1, ζ, ζ 2, ..., ζ p−2). If d {\displaystyle d} is a square-free integer and K = Q ( d ) {\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)} is
Ring_of_integers
Product of an integer with itself
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 itself
Square_number
Mathematical concept
{\displaystyle {a+b{\sqrt {c}} \over d},} for integers a, b, c, d; with b, c and d non-zero, and with c square-free. When c is positive, we get real quadratic
Quadratic_irrational_number
Complex number that solves a monic polynomial with integer coefficients
{n}}} of a nonnegative integer n is an algebraic integer, but is irrational unless n is a perfect square. If d is a square-free integer then the extension
Algebraic_integer
Field (mathematics) generated by the square root of an integer
({\sqrt {d}})} where d {\displaystyle d} is a (uniquely defined) square-free integer different from 0 {\displaystyle 0} and 1 {\displaystyle 1} . If d
Quadratic_field
Natural number
57 and preceding 59. 58 is a composite number, a semiprime, and a square-free integer. 58 is the sum of the first seven primes. Sloane, Neil; Guy, R. K
58_(number)
Area of a right triangle with rational-numbered sides
group contains exactly one square-free integer, and it is common, therefore, only to consider square-free positive integers when speaking about congruent
Congruent_number
Number in {..., –2, –1, 0, 1, 2, ...}
example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, 5/4, and the square root of 2 are not. The integers form the smallest group and the smallest
Integer
Product of the prime factors of an integer
radical of any integer n {\displaystyle n} is the largest square-free divisor of n {\displaystyle n} and so also described as the square-free kernel of n
Radical_of_an_integer
Natural number
sum of 427 when excluding a square of side length 7 {\displaystyle 7} ; this sum represents the largest square-free integer over a quadratic field of class
21_(number)
On areas of rational right triangles
solutions of a few fairly simple Diophantine equations. For a given square-free integer n, define A n = # { ( x , y , z ) ∈ Z 3 ∣ n = 2 x 2 + y 2 + 32 z
Tunnell's_theorem
Natural number
and 71. 69 is not divisible by any square number other than 1, making it a square-free integer. 69 is a Blum integer since the two factors of 69 are both
69_(number)
Decomposition of an integer as a sum of positive integers
partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only
Integer_partition
Generalizations of prime ideals and prime rings
of integers, the semiprime ideals are the zero ideal, along with those ideals of the form n Z {\displaystyle n\mathbb {Z} } where n is a square-free integer
Semiprime_ring
Mathematical group
positive square-free integer. Here, PSL denotes the projective special linear group and O d {\displaystyle {\mathcal {O}}_{d}} is the ring of integers of the
Bianchi_group
would make it divisible by a square number). Common examples of square-free elements include square-free integers and square-free polynomials. Prime number
Square-free_element
A square-free integer is an integer that is not divisible by any square other than 1. square number A square number is an integer that is the square of
Glossary_of_number_theory
Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power of two Integer-valued polynomial Rational number
List_of_number_theory_topics
Unproved conjecture in mathematics
conjecture has multiple consequences, including: Let n be an odd square-free integer. Assuming the Birch and Swinnerton-Dyer conjecture, n is the area
Birch and Swinnerton-Dyer conjecture
Birch_and_Swinnerton-Dyer_conjecture
Natural number
Sloane, N. J. A. (ed.). "Sequence A001110 (Square triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A
1,000,000
Decomposition of a number into a product
decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater
Integer_factorization
Algorithm for generating pseudo-randomized numbers
using a computer's word size is the most common choice. If m were a square-free integer, this would only allow a ≡ 1 (mod m), which makes a very poor PRNG;
Linear_congruential_generator
Function of the coefficients of a polynomial that gives information on its roots
and is square-free, meaning it is not divisible by the square of any prime number. Case 2: D {\textstyle D} is equal to four times an integer m {\textstyle
Discriminant
In number theory, measure of non-unique factorization
countably infinite set of ideal classes. If d {\displaystyle d} is a square-free integer (a product of distinct primes) other than 1, then Q ( d ) {\displaystyle
Ideal_class_group
Natural number
the sum of three cubes. There are nine Heegner numbers, or square-free positive integers n {\displaystyle n} that yield an imaginary quadratic field
9
Online database of integer sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Root-finding algorithm
approximation through integer operations by adding and subtracting the integer form of floating-point numbers, and taking a square root by dividing by two
Fast_inverse_square_root
This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. OEIS core sequences Index to
List_of_integer_sequences
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A000105 (Number of free polyominoes (or square animals) with n cells)"
1,000,000,000,000
Function in mathematical number theory
positive integer n is the smallest positive integer m such that a m ≡ 1 ( mod n ) {\displaystyle a^{m}\equiv 1{\pmod {n}}} holds for every integer a coprime
Carmichael_function
Natural number
square numbers in two distinct ways. 50 is an unsigned Stirling number of the first kind and a Narayana number. Look up fifty in Wiktionary, the free
50_(number)
Natural number
(Numbers k such that the sum of the squares of the first k primes is divisible by k)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane
100,000,000
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
Theorem in number theory
Every n in Mx can be written as n = m2r with positive integers m and r, where r is square-free. Since only the k primes p1, ..., pk can show up (with
Divergence of the sum of the reciprocals of the primes
Divergence_of_the_sum_of_the_reciprocals_of_the_primes
Quadratic imaginary number fields with unique factorisation
number. Let Q denote the set of rational numbers, and let d be a square-free integer. The field Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}})} is a
Stark–Heegner_theorem
Algebraic curve in mathematics
including the following two: A congruent number is defined as an odd square-free integer n which is the area of a right triangle with rational side lengths
Elliptic_curve
Numbers whose differences are not squares
construction chooses a square-free integer b {\displaystyle b} as the radix of the base- b {\displaystyle b} notation for the integers, such that there exists
Square-difference-free_set
Concept in field theory mathematics
\mathbb {Q} ({\sqrt {a}})/\mathbb {Q} } where a {\displaystyle a} is a square-free integer. Then, the multiplication map by a {\displaystyle {\sqrt {a}}} on
Field_norm
Natural number
American English, its name has a semiprime number of syllables. It is a square-free integer. It is a nontotient. It is a lazy caterer number. It is a congruent
254_(number)
that Q has class number 1. These are of the form K = Q(√d), for a square-free integer d. K is called real quadratic if d > 0. K has class number 1 for
List of number fields with class number one
List_of_number_fields_with_class_number_one
Sum type in number theory
geometric sum formula then shows that G(a, b, 2m) = 0. If c is an odd square-free integer and gcd(a, c) = 1, then G ( a , 0 , c ) = ∑ n = 0 c − 1 ( n c ) e
Quadratic_Gauss_sum
Natural number
J. A. (ed.). "Sequence A112643 (Odd and square-free abundant numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. "A002182 - OEIS"
10,000
Unique positive real number which when multiplied by itself gives 2
A002193 in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 30 decimal
Square_root_of_2
Natural number
positive integer that can be expressed as the difference of two positive squares in at least n ways.)". The On-Line Encyclopedia of Integer Sequences
15_(number)
Symbol in number theory
F=\mathbb {Q} ({\sqrt {m}})} , where m {\displaystyle m} is a nonzero square-free integer (we can include the case Q ( 1 ) = Q {\displaystyle \mathbb {Q} ({\sqrt
Kronecker_symbol
Natural number
A122400 (Number of square (0,1)-matrices without zero rows and with exactly n entries equal to 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
10,000,000
Natural number
yields primes for all the integers k with 1 ≤ k < 41. the sum of two consecutive squares (42 + 52), which makes it a centered square number. the sum of the
41_(number)
{\displaystyle K=\mathbf {Q} ({\sqrt {d}})} with d {\displaystyle d} a square-free integer, then O K = Z [ a ] {\displaystyle O_{K}=\mathbf {Z} [a]} where a
Monogenic_field
Natural number
Furthermore, 290 is the sum of the squares of divisors of 17, or 289 + 1. "Sloane's A060315". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
29_(number)
Natural number
of "4". Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four squares. Four is one of four all-Harshad
4
Natural number
apocalyptic power (21510 contains the consecutive digits 666). 1510 is a square-free integer. 1510 is an untouchable number. 1510 is a subprime number (1510 +
1510_(number)
Natural number, composite number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-18. Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers)".
14_(number)
Discrete group of Möbius transformations
ring of integers of the imaginary quadratic field Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} for d a positive square-free integer. A Kleinian
Kleinian_group
Natural number
centered cube number. a square pyramidal number, being the sum of the squares of the first six integers. the smallest positive integer expressible as a sum
91_(number)
Computational method
{\displaystyle f(x)} is a univariate polynomial over the integers, assumed to be content-free and square-free, one starts by computing a bound B {\displaystyle
Factorization_of_polynomials
) , {\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,),} where d is a square-free integer, and that the narrow class group of K is trivial. Suppose that {
Narrow_class_group
Natural number
highly composite number, and the first colossally abundant number. An integer is determined to be even if it is divisible by two. When written in base
2
partitions of 1 into reciprocals of positive integers <= 17 Egyptian fraction 1013 = Sophie Germain prime, centered square number, Mertens function zero 1014 =
1000_(number)
Summatory function of the Möbius function
).} Less formally, M ( x ) {\displaystyle M(x)} is the count of square-free integers up to x that have an even number of prime factors, minus the count
Mertens_function
Finite extension of the rationals
space over Q {\displaystyle \mathbb {Q} } . More generally, for any square-free integer d {\displaystyle d} , the quadratic field Q ( d ) {\displaystyle
Algebraic_number_field
Concept in number theory
are more positive integers than perfect squares, because every perfect square is already positive and yet many other positive integers exist besides. However
Natural_density
Object from algebraic number theory
} Without loss of generality, we may assume a and b are non-zero square-free integers. The Galois group has three subgroups of index 2, corresponding to
Biquadratic_field
500, 648, 675, 800, 864, 968 (sequence A052486 in the OEIS). A square-free integer has no prime factor with multiplicity greater than 1. The first:
Table_of_prime_factors
Natural number
of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A001844 (Centered square numbers)". The On-Line Encyclopedia of Integer Sequences
4000_(number)
Used to count, measure, and label
Hindu–Arabic numeral system, a decimal system which can display any non-negative integer using a combination of ten Arabic numeral symbols called digits. Numerals
Number
Integer side lengths of a right triangle
triple because the square root of 2 is not an integer. Moreover, 1 {\displaystyle 1} and 2 {\displaystyle {\sqrt {2}}} do not have an integer common multiple
Pythagorean_triple
Multiset of prime exponents in a prime factorization
number if S = {1}, A square if gcd(S) is even, A cube if gcd(S) is divisible by 3, A square-free integer if max(S) = 1, A cube-free integer if max(S) ≤ 2, A
Prime_signature
Natural number
numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers: a(n) = 0^2 +
100,000
Natural number
proved that every integer is the sum of at most three triangular numbers. Three is the only prime which is one less than a perfect square. Any other number
3
Product of a number by itself
to squaring is quadratic. The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often
Square_(algebra)
Natural number
lowest natural number that cannot be represented as the sum of the squares of three integers. A seven-sided shape is a heptagon. The regular n-gons for n ⩽
7
Number of prime factors of a natural number
function Erdős–Kac theorem Omega function (disambiguation) Prime number Square-free integer This inequality is given in Section 22.13 of Hardy and Wright. S
Prime_omega_function
Two consecutive integers for which the sums of the prime factors of each are equal
pair of square-free integers belongs to both lists with the same sum of prime factors. The intersection also contains pairs that are not square-free, for
Ruth–Aaron_pair
Integer that divides another integer
mathematics, a divisor of an integer n , {\displaystyle n,} also called a factor of n , {\displaystyle n,} is an integer m {\displaystyle m} that may
Divisor
Arithmetic function related to the divisors of an integer
related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number
Divisor_function
Natural number
The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A016754 (Odd squares: a(n) = (2n+1)^2. Also centered
81_(number)
Wiki-based programming chrestomathy
SEDOLs Semiprimes Sierpinski triangle (draw) Sorting algorithms (41) Square-free integers Statistics Stem-and-leaf display Function definition Sudoku (solve)
Rosetta_Code
Replacing a number with a simpler value
especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or
Rounding
Natural number
base 36. 1,631,432,881 = 403912, square triangular number 1,673,196,525 : Least common multiple of the odd integers from 1 to 25 1,677,922,740 : number
1,000,000,000
Number used for counting
2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
Natural number
four positive integers (100 = 13 + 23 + 33 + 43). This is related by Nicomachus's theorem to the fact that 100 also equals the square of the sum of the
100
Measures the size of the ring of integers of the algebraic number field
measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume of the fundamental domain
Discriminant of an algebraic number field
Discriminant_of_an_algebraic_number_field
Natural number
Theaetetus, it is believed that Theodorus had proved all the square roots of non-square integers from 3 to 17 are irrational by means of this spiral. In three-dimensional
17_(number)
Natural number
× 4 matrices with nonnegative integer entries and row and column sums equal to 3 2009 = 282 + 352, sum of two squares 2010 – number of compositions of
2000_(number)
Natural number between 89 and 91
(6^{2}+5^{2}+4^{2}+3^{2}+2^{2})} . The square of eleven 112 = 121 is the ninetieth indexed composite number, where the sum of integers { 2 , 3 , . . . , 11 } {\displaystyle
90_(number)
In combinatorics, a square-free word is a word (a sequence of symbols) that does not contain any squares. A square is a word of the form XX, where X is
Square-free_word
Method to solve optimization problems
case, integer programming problems are in many practical situations (those with bounded variables) NP-hard. 0–1 integer programming or binary integer programming
Linear_programming
Natural number
of the meaning of time. "Sloane's A002110 : Primorial numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 31 May 2016.
30_(number)
Algebraic structure
rings are free commutative algebras, since they are free objects in the category of commutative algebras. Similarly, a polynomial ring with integer coefficients
Polynomial_ring
Mathematical concept
are related to special values of Euler products". For every square-free positive integer n pick an n-th root ζn of 1, with ζmn = ζmζn for m,n coprime
Euler_system
Mathematical structure with greatest common divisors
{\displaystyle G} is torsion-free. The ring Z [ − d ] {\displaystyle \mathbb {Z} [{\sqrt {-d}}]} is not a GCD domain for all square-free integers d ≥ 3 {\displaystyle
GCD_domain
Natural number
than there are for any other integers below it except 1. There are 35 free hexominoes, the polyominoes made from six squares. Since the greatest prime factor
35_(number)
Natural number
divisors being 1, 2, 4 and 8. Sixteen is the only integer that equals mn and nm, for some unequal integers m and n ( m = 4 {\displaystyle m=4} , n = 2 {\displaystyle
16_(number)
Topics referred to by the same term
Square number, an integer that is the square of another integer Square of a graph Square wave (waveform), a non-sinusoidal periodic waveform Square (area)
Square_(disambiguation)
Natural number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved June 2, 2022. Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers)"
500_(number)
Natural number
Look up fifty-four in Wiktionary, the free dictionary. 54 (fifty-four) is the natural number and positive integer following 53 and preceding 55. As a multiple
54_(number)
expressed as a ratio of an integer to a non-zero integer. All integers are rational, but there are rational numbers that are not integers, such as −2/9. Real
List_of_types_of_numbers
Natural number
number, as the solution to x − ϕ ( x ) {\displaystyle x-\phi (x)} for the integers 95, 119, 143, and 529. 23 is the second Smarandache–Wellin prime in base
23_(number)
SQUARE FREE-INTEGER
SQUARE FREE-INTEGER
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Surname or Lastname
English
English : from Old French and Middle English frere ‘friar’ (Latin frater, literally ‘brother’). This was a status name for a member a religious order, especially a mendicant order, and may also have been a nickname for a pious person or for someone employed at a monastery.Americanized spelling of French Frère (see Frere).North German and Dutch : cognate of Friedrich.
Boy/Male
English American
Shieldbearer.
Surname or Lastname
English (chiefly East Anglia)
English (chiefly East Anglia) : nickname or status name from Old English frēo ‘free(-born)’, i.e. not a serf.North German : topographic or habitational name from a place named Frede or Frede(n).North German : nickname from a variant of Middle Low German wrēd ‘crooked’.
Male
Swedish
Danish and Swedish form of Old Norse Freyr, FREJ means "lord, master."
Surname or Lastname
English
English : variant of Freer 1.French (Frère) : from frère ‘brother’, used as a byname for the younger of two brothers.
Surname or Lastname
English
English : patronymic from Squire.
Surname or Lastname
English
English : nickname from Middle English freil, frel(i)e ‘frail’, ‘weak’.Possibly an Americanized spelling of German Friel 2.
Male
English
Short form of English Frederick, FRED means "peaceful ruler."
Boy/Male
Italian
Squire.
Surname or Lastname
English
English : variant of Squire.
Female
English
Anglicized form of Danish Freya, FREA means "lady, mistress."
Male
Swedish
Swedish name derived from Old Norse stúra, STURE means "obstinate."
Boy/Male
American, Australian, British, English
Shield Bearer; Knight's Companion
Male
Icelandic
Icelandic form of Old Norse Freyr, FREY means "lord, master."
Surname or Lastname
English (mainly southeastern)
English (mainly southeastern) : topographic name for someone who lived near a conspicuous tree, Middle English tre(w).
Female
English
English form of Irish BrÃgh, BREE means "force, strength."
Male
English
French form of English Stewart, STUART means "house guard; steward." In use by the English and Scottish.
Boy/Male
Australian, British, English
In Liberty
SQUARE FREE-INTEGER
SQUARE FREE-INTEGER
Female
English
Feminine form of Latin Felix, FELICIA means "happy" or "lucky."
Girl/Female
Hindu, Indian
Star
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Sindhi, Telugu
Supreme Consciousness
Girl/Female
Australian, French, Latin, Spanish
Born on Christmas; Birthday; Reference to the Nativity; Christmas; Form of Natalie
Girl/Female
Hindu, Indian
Veda
Boy/Male
Hindu
Fought all the battle by fair means
Girl/Female
Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Goddess Durga
Female
English
Variant spelling of English Janine, JANNINE means "God is gracious."
Female
English
English variant form of Celtic Boudica, BONDUCA means "victory."
Female
English
English variant form of Spanish Anita, ANETTA means "favor; grace."
SQUARE FREE-INTEGER
SQUARE FREE-INTEGER
SQUARE FREE-INTEGER
SQUARE FREE-INTEGER
SQUARE FREE-INTEGER
n.
To multiply by itself; as, to square a number or a quantity.
imp. & p. p.
of Square
n.
Having the toe square.
a.
Even; leaving no balance; as, to make or leave the accounts square.
n.
A square; a measure; a rule.
n.
A square. See 1st Squire.
superl.
Privileged or individual; the opposite of common; as, a free fishery; a free warren.
a.
Having four equal sides and four right angles; as, a square figure.
a.
Of or pertaining to a square, or to squares; resembling a quadrate, or square; square.
a.
Forming a right angle; as, a square corner.
adv.
Without charge; as, children admitted free.
n.
Hence, anything which is square, or nearly so
n.
An instrument having at least one right angle and two or more straight edges, used to lay out or test square work. It is of several forms, as the T square, the carpenter's square, the try-square., etc.
a.
Rendering equal justice; exact; fair; honest, as square dealing.
superl.
Certain or honorable; the opposite of base; as, free service; free socage.
n.
To place at right angles with the keel; as, to square the yards.
superl.
Not gained by importunity or purchase; gratuitous; spontaneous; as, free admission; a free gift.
n.
The product of a number or quantity multiplied by itself; thus, 64 is the square of 8, for 8 / 8 = 64; the square of a + b is a2 + 2ab + b2.
n.
A square piece or fragment.
n.
One who frees, or sets free.