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Arithmetic function related to the divisors of an integer
number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the
Divisor_function
Summatory function of the divisor-counting function
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic
Divisor_summatory_function
Largest integer that divides given integers
In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the
Greatest_common_divisor
Integer that divides another integer
In 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
Function definition that is not bound to an identifier
functions with a specified divisor. The functions half and third curry the divide function with a fixed divisor. The divisor function also forms a closure by
Anonymous_function
Certain type of divisor of an integer
mathematics, a natural number a is a unitary divisor (or Hall divisor) of a number b if a is a divisor of b and if a and b / a are coprime, having no
Unitary_divisor
Generalizations of codimension-1 subvarieties of algebraic varieties
divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors
Divisor_(algebraic_geometry)
Number divisible only by 1 and itself
number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are for 1.
Prime_number
Number of integers coprime to and less than n
, and which have no common divisor with it". This definition varies from the current definition for the totient function at D = 1 {\displaystyle D=1}
Euler's_totient_function
Nineteenth letter in the Greek alphabet
and chronic traumatic encephalopathy Divisor function in number theory, also denoted d or σ0 Ramanujan tau function Golden ratio (1.618...), although φ
Tau
necessarily also a divisor of n). For example, 3 is a divisor of 21, since 21/7 = 3 (and therefore 7 is also a divisor of 21). If m is a divisor of n, then
Table_of_divisors
Numbers obtained by adding the two previous ones
\ldots )=F_{\gcd(a,b,c,\ldots )}\,} where gcd is the greatest common divisor function. (This relation is different if a different indexing convention is
Fibonacci_sequence
Positive integer whose divisors have a harmonic mean that is an integer
harmonic 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
Harmonic_divisor_number
Integer having a non-trivial divisor
factorization Sieve of Eratosthenes Table of prime factors Divisor function Prime omega function Möbius function Pettofrezzo & Byrkit 1970, pp. 23–24. Long 1972
Composite_number
Class of natural numbers with many divisors
the divisor function, denotes the number of divisors of n. The term was coined by Ramanujan (1915). For example, the number with the most divisors per
Superior highly composite number
Superior_highly_composite_number
Function whose domain is the positive integers
prime-counting functions. This article provides links to functions of both classes. An example of an arithmetic function is the divisor function whose value
Arithmetic_function
Function equal to the product of its values on coprime factors
{\displaystyle \sigma _{k}(n)} : the divisor function, which is the sum of the k {\displaystyle k} -th powers of all the positive divisors of n {\displaystyle n} (where
Multiplicative_function
Integer which is the sum of its positive unitary divisors, not including itself
of its positive proper unitary divisors, not including the number itself. (A divisor d of a number n is a unitary divisor if d and n/d share no common factors
Unitary_perfect_number
Type of Poulet number
every divisor d {\displaystyle d} divides 2 d − 2 {\displaystyle 2^{d}-2} . For example, 341 is a super-Poulet number: it has positive divisors (1, 11
Super-Poulet_number
Pair of integers related by their divisors
itself (see also divisor function). The smallest pair of amicable numbers is (220, 284). They are amicable because the proper divisors of 220 are 1, 2
Amicable_numbers
Number equal to the sum of its proper divisors
positive divisors; in symbols, σ 1 ( n ) = 2 n {\displaystyle \sigma _{1}(n)=2n} where σ 1 {\displaystyle \sigma _{1}} is the sum-of-divisors function. This
Perfect_number
Class of mathematical functions
d m {\displaystyle \sigma _{m}(k):=\sum _{d\mid {k}}d^{m}} is the divisor function and q = e π i τ {\displaystyle q=e^{\pi i\tau }} is the nome. The modular
Weierstrass_elliptic_function
Number that cannot be written as an aliquot sum
sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function. Their study goes back
Untouchable_number
Formal power series
where we have the special case identity for the generating function of the divisor function, d(n) ≡ σ0(n), given by ∑ n = 1 ∞ x n 1 − x n = ∑ n = 1 ∞ x
Generating_function
Topics referred to by the same term
by sigma function one can mean one of the following: The sum-of-divisors function σa(n), an arithmetic function Weierstrass sigma function, related to
Sigma_function
Topics referred to by the same term
coefficients of the Ramanujan modular form Divisor function, an arithmetic function giving the number of divisors of an integer This disambiguation page lists
Tau_function
Number used for counting
numbers a, b, and c, a × (b + c) = (a × b) + (a × c). No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b =
Natural_number
Function studied by Ramanujan
\mathbb {N} } , the divisor function σ k ( n ) {\displaystyle \sigma _{k}(n)} is the sum of the k {\displaystyle k} th powers of the divisors of n {\displaystyle
Ramanujan_tau_function
Rule for proportional allocation
The highest averages, divisor, or divide-and-round methods are a family of apportionment rules, i.e. algorithms for fair division of seats in a legislature
Highest_averages_method
Sum of all proper divisors of a natural number
sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself. That is, s ( n ) = ∑ d | n , d
Aliquot_sum
Number of prime factors of a natural number
constants. The function ω ( n ) {\displaystyle \omega (n)} is related to divisor sums over the Möbius function and the divisor function, including: ∑ d
Prime_omega_function
Recursive integer sequence
binomial coefficients, by Stirling's approximation for n!, or via generating functions. The only Catalan numbers Cn that are odd are those for which n = 2k −
Catalan_number
Mathematical function
Hooley's delta function ( Δ ( n ) {\displaystyle \Delta (n)} ), also called Erdős--Hooley delta-function, defines the maximum number of divisors of n {\displaystyle
Hooley's_delta_function
Type of natural number
{\frac {\sigma (k)}{k^{1+\varepsilon }}}} where σ denotes the sum-of-divisors function. The first 15 colossally abundant numbers, 2, 6, 12, 60, 120, 360
Colossally_abundant_number
Figurate number
with the factorial function, a product whose factors are the integers from 1 to n, Donald Knuth proposed the name Termial function, with the notation
Triangular_number
Mathematical operation on arithmetical functions
mathematics, Dirichlet convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory. It
Dirichlet_convolution
Number with a half-integer abundancy index
k/2 for an odd integer k, where σ(n) is the sum-of-divisors function, the sum of all positive divisors of n. The first few hemiperfect numbers are: 2, 24
Hemiperfect_number
Number-theoretical function
one can express r 4 ( n ) {\displaystyle r_{4}(n)} in terms of the divisor function as follows: r 4 ( n ) = 8 σ ( 2 min { k , 1 } m ) . {\displaystyle
Sum_of_squares_function
Number whose divisors add to a multiple of that number
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 only
Multiply_perfect_number
Number equal to the sum of all or some of its divisors
the sum of all or some of its proper divisors. A semiperfect number equal to the sum of all its proper divisors is a perfect number. The first few semiperfect
Semiperfect_number
{\displaystyle \tau } is the divisor function, and μ {\displaystyle \mu } is the Möbius function. This multiplicative arithmetical function was introduced by the
Pillai's arithmetical function
Pillai's_arithmetical_function
Number that is less than the sum of its proper divisors
which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for
Abundant_number
Positive integer that is an integer power of another positive integer
values for k across each of the divisors of n, up to k ≤ log 2 n {\displaystyle k\leq \log _{2}n} . So if the divisors of n {\displaystyle n} are n 1
Perfect_power
Number that is abundant but not semiperfect
of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number
Weird_number
Two or more natural numbers with a common abundancy index
\sigma _{k}} denotes a divisor function with σ k ( n ) {\displaystyle \sigma _{k}(n)} equal to the sum of the k-th powers of the divisors of n. The numbers
Friendly_number
Computational operation
\rfloor } is the floor function (rounding down). Thus according to equation (1), the remainder has the same sign as the divisor n: r = a − n ⌊ a n ⌋ {\displaystyle
Modulo
Integer sequence in number theory
if only 22021 were a prime number, since in that case the sum-of-divisors function for D would satisfy σ ( D ) = ( 3 2 + 3 + 1 ) ⋅ ( 7 2 + 7 + 1 ) ⋅
Descartes_number
Number of partitions of an integer
p ( n ) {\displaystyle p(n)} can be given in terms of the sum of divisors function σ: p ( n ) = 1 n ∑ k = 0 n − 1 σ ( n − k ) p ( k ) . {\displaystyle
Partition function (number theory)
Partition_function_(number_theory)
Numbers that evenly divide powers of 60
they are the numbers whose only prime divisors are 2, 3, and 5. As an example, 602 = 3600 = 48 × 75, so as divisors of a power of 60 both 48 and 75 are
Regular_number
Eighteenth letter of the Greek alphabet
number theory, σ is included in various divisor functions, especially the sigma function or sum-of-divisors function. In applied mathematics, σ(T) denotes
Sigma
Positive integer that is the product of three distinct prime numbers
exactly eight divisors. All sphenic numbers are by definition squarefree, because the prime factors must be distinct. The Möbius function of any sphenic
Sphenic_number
Integers have unique prime factorizations
numbers. The canonical representations of the product, greatest common divisor (GCD), and least common multiple (LCM) of two numbers a and b can be expressed
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Iterative algorithm on numbers
sequence. Repeat step 2. The sequence is called a Kaprekar sequence and the function K b ( n ) = α − β {\displaystyle K_{b}(n)=\alpha -\beta } is the Kaprekar
Kaprekar's_routine
Product of two prime numbers
where π ( x ) {\displaystyle \pi (x)} is the prime-counting function and p k {\displaystyle p_{k}} denotes the kth prime. Semiprime numbers
Semiprime
Base-dependent property of integers
(d)=d\ {\text{Inv}}(d,e)} . Then the function ζ {\displaystyle \zeta } is a bijection from the set of unitary divisors of N − 1 {\displaystyle N-1} onto
Kaprekar_number
Arithmetic operation
theory. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the order of the group). The possible
Exponentiation
Number without repeated prime factors
has no t-th power in its divisors. In particular, the 2-free integers are the square-free integers. The multiplicative function c o r e t ( n ) {\displaystyle
Square-free_integer
Decomposition of a number into a product
order dividing 2 to obtain a coprime factorization of the largest odd divisor of Δ in which Δ = −4ac or Δ = a(a − 4c) or Δ = (b − 2a)(b + 2a). If the
Integer_factorization
Summatory function of the Möbius function
the Mertens function suggests asymptotic bounds obtained by considering the Piltz divisor problem, which generalizes the Dirichlet divisor problem of computing
Mertens_function
Integer divisible by the number of its divisors
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 ) ∣ n {\displaystyle
Refactorable_number
Numbers whose prime factors all divide the number more than once
X(49Y3 + 81Z3), Y′ = −Y(32X3 + 81Z3), Z′ = Z(32X3 − 49Y3) and omitting the common divisor. Achilles number Highly powerful number "Squarefull numbers". OEIS Wiki
Powerful_number
useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number n {\displaystyle n}
Divisor_sum_identities
Count of the possible partitions of a set
period of this repetition, for an arbitrary prime number p, must be a divisor of p p − 1 p − 1 {\displaystyle {\frac {p^{p}-1}{p-1}}} and for all prime
Bell_number
Type of figurate number
distinct divisor, so r n − 1 {\displaystyle r^{n-1}} has [ 2 ( n − 1 ) + 1 ] [ ( n − 1 ) + 1 ] {\displaystyle [2(n-1)+1][(n-1)+1]} divisors, i.e. ( 2
Hexagonal_number
Product of an integer with itself
number of positive divisors, while other natural numbers have an even number of positive divisors. An integer root is the only divisor that pairs up with
Square_number
Integer having only small prime factors
n-powersmooth numbers. In fact, the n-powersmooth numbers are exactly the positive divisors of “the least common multiple of 1, 2, 3, …, n” (sequence A003418 in the
Smooth_number
Divergent sum of positive unit fractions
average number of divisors of the numbers in a range from 1 to n {\displaystyle n} , formalized as the average order of the divisor function, 1 n ∑ i = 1 n
Harmonic_series_(mathematics)
Number that represents a hexagon with a dot in the center
calculate the generating function F ( x ) = ∑ n ≥ 0 H ( n ) x n {\displaystyle F(x)=\sum _{n\geq 0}H(n)x^{n}} . The generating function satisfies F ( x ) =
Centered_hexagonal_number
Number whose divisors summed twice over equal twice itself
{\displaystyle \sigma ^{2}(n)=\sigma (\sigma (n))=2n\,,} where σ is the sum-of-divisors function. Superperfect numbers are not a generalization of perfect numbers
Superperfect_number
Number whose sums of distinct divisors represent all smaller numbers
divisors of n {\displaystyle n} . For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors
Practical_number
Relation between pairs of arithmetic functions
formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832
Möbius_inversion_formula
Abundant number whose proper divisors are all deficient numbers
whose proper divisors are all deficient numbers. For example, 20 is a primitive abundant number because: The sum of its proper divisors is 1 + 2 + 4 +
Primitive_abundant_number
Number, product of consecutive integers
Divisibility-based sets of integers Overview Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Factorization
Pronic_number
Numbers whose sum of divisors is twice the number plus 1
number is a natural number n for which the sum of all its divisors (the sum-of-divisors function σ ( n ) {\displaystyle \sigma (n)} ) is equal to 2 n + 1
Quasiperfect_number
Concept in number theory
(b^{2}+1)-2} , where τ ( n ) {\displaystyle \tau (n)} is the number of positive divisors of n {\displaystyle n} . Every base b ≥ 3 {\displaystyle b\geq 3} that
Narcissistic_number
Mathematical recursive sequence
integer k can be defined formally in terms of the sum-of-divisors function σ1 or the aliquot sum function s in the following way: s 0 = k s n = s ( s n − 1 )
Aliquot_sequence
Difference between logarithm and harmonic series
algorithm. Sums involving the Möbius and von Mangolt function. Estimate of the divisor summatory function of the Dirichlet hyperbola method. In some formulations
Euler's_constant
Integer where the average of its positive divisors is also an integer
average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is 1 + 2 + 3 + 6 4 = 3
Arithmetic_number
Arithmetic function
putting both g = h = 1, where 1(n) = 1 is the constant function. Here τ is the divisor function. f ⋅ ( g ∗ h ) ( n ) = f ( n ) ⋅ ∑ d | n g ( d ) h ( n
Completely multiplicative function
Completely_multiplicative_function
Natural number whose divisor sum is greater than that of any smaller number
{\displaystyle \sigma (n)>\sigma (m)} where σ denotes the sum-of-divisors function. The first few highly abundant numbers are 1, 2, 3, 4, 6, 8, 10, 12
Highly_abundant_number
Number that has fewer digits than the number of digits in its prime factorization
Divisibility-based sets of integers Overview Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Factorization
Extravagant_number
Relation between genus, degree, and dimension of function spaces over surfaces
surface with integer coefficients. Any meromorphic function f {\displaystyle f} gives rise to a divisor denoted ( f ) {\displaystyle (f)} defined as ( f
Riemann–Roch_theorem
fixed points. A000166 Divisor function σ(n) 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, ... σ(n) := σ1(n) is the sum of divisors of a positive integer n
List_of_integer_sequences
Conjecture on zeros of the zeta function
disproving) it. Some typical examples are as follows. (Others involve the divisor function σ(n).) The Riesz criterion was given by Riesz (1916), to the effect
Riemann_hypothesis
Class of natural numbers
{\sigma (n)}{n}}} where σ denotes the sum-of-divisors function (i.e., the sum of all positive divisors of n, including n itself). The first few superabundant
Superabundant_number
Multiplicative function Additive function Dirichlet convolution Erdős–Kac theorem Möbius function Möbius inversion formula Divisor function Liouville function Partition
List_of_number_theory_topics
Count of permutations by cycles
, v ) {\displaystyle \zeta (k,v)} are the Riemann zeta function and the Hurwitz zeta function respectively, and even evaluate this integral ∫ 0 1 log
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
Mathematics
set of three different numbers so related that the restricted sum of the divisors of each is equal to the sum of other two numbers. In another equivalent
Amicable_triple
Type of number introduced by Mike Keith
Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally
Keith_number
Infinite integer series where the next number is the sum of the two preceding it
− 4 ( 18 ) + 6 {\displaystyle 256=322-4(18)+6} The ordinary generating function of the sequence of Lucas numbers is the power series Φ ( x ) = ∑ k = 0
Lucas_number
Mathematical term
_{0}(n)=d(n)} is the number of positive divisors of the number n. For the higher order sum-of-divisor functions, one has ∑ n = 1 ∞ q n σ α ( n ) = ∑ n
Lambert_series
Numbers with a certain property involving recursive summation
eventually reaches 1 when iterated over the perfect digital invariant function for p = 2 {\displaystyle p=2} . The origin of happy numbers is not clear
Happy_number
Number that remains the same when its digits are reversed
Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally
Palindromic_number
Function in number theory given by Srinivasa Ramanujan
greatest common divisor, ϕ ( n ) {\displaystyle \phi (n)} is Euler's totient function, μ ( n ) {\displaystyle \mu (n)} is the Möbius function, and ζ ( s )
Ramanujan's_sum
constant function g ( x ) = c {\displaystyle g(x)=c} is an average order of f {\displaystyle f} . An average order of d(n), the number of divisors of n,
Average order of an arithmetic function
Average_order_of_an_arithmetic_function
Number that has a perfect number of factors adding up to another perfect number
Divisibility-based sets of integers Overview Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Factorization
Sublime_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
Integer whose representation contains every digit in its number base
divisor of any pandigital number is 111,119. However, some larger prime numbers do occur as divisors of pandigital numbers. The largest prime divisor
Pandigital_number
Type of positive integer pairs
condition is that σ(m) = σ(n) = m + n + 1, where σ denotes the sum-of-divisors function. The first few pairs of betrothed numbers are: (48, 75), (140, 195)
Betrothed_numbers
Integer filtered out using a sieve similar to that of Eratosthenes
Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally
Lucky_number
DIVISOR FUNCTION
DIVISOR FUNCTION
Boy/Male
Muslim
Advisor
Girl/Female
Arabic, Muslim, Sindhi
Advisor
Girl/Female
Muslim
Advisor
Boy/Male
Muslim
Advisor.
Girl/Female
Hindu, Indian, Tamil
Devine Wish
Boy/Male
Muslim
Advisor
Girl/Female
Biblical
Division.
Boy/Male
Muslim
Divider.
Boy/Male
American, Australian, British, English
Son of David; David's Son; Surname
Boy/Male
English
David's son. Surname.
Girl/Female
Indian
Goddess Durga, Chief of the Goddess, Devee
Boy/Male
Indian
Advisor
Boy/Male
Indian
Advisor
Boy/Male
Muslim
Advisor.
Boy/Male
French, German
Honest Advisor; Brave Advisor
Boy/Male
Hindu, Indian
Beautiful Sky
Boy/Male
Indian
Advisor
Boy/Male
Muslim
Advisor
Boy/Male
Indian
Divider
Girl/Female
Biblical
Division.
DIVISOR FUNCTION
DIVISOR FUNCTION
Girl/Female
American, Anglo, British, English
Noble Strength; Nobility
Boy/Male
Indian, Sanskrit
Affection; Desire
Boy/Male
British, Danish, English, Finnish, German, Scandinavian, Swedish
People's Guardian; Chief; Stranger Man; Army; Warrior
Boy/Male
Hindu
The Lord Vishnu
Girl/Female
Bengali, Hindu, Indian, Kannada, Marathi
Full of Grace and Lily
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : habitational name from either of two places in West Yorkshire now called Holdsworth, both probably originally named with an Old English byname Halda ‘bent’ + worð ‘enclosure’.
Girl/Female
Tamil
Likhitha | லிகித , லீகீதா , லீகீதாÂ
Writing
Girl/Female
Arabic Muslim
Conversations at night.
Girl/Female
Hindu, Indian, Marathi
The Fair Girl
Boy/Male
Muslim
Servant of the one
DIVISOR FUNCTION
DIVISOR FUNCTION
DIVISOR FUNCTION
DIVISOR FUNCTION
DIVISOR FUNCTION
n.
The distribution of a discourse into parts; a part so distinguished.
n.
A conjecture; a guesser; one who makes out occult things.
n.
Separation of the members of a deliberative body, esp. of the Houses of Parliament, to ascertain the vote.
n.
That which divides or keeps apart; a partition.
n.
The process of finding how many times one number or quantity is contained in another; the reverse of multiplication; also, the rule by which the operation is performed.
n.
The number by which the dividend is divided.
n.
A soldier who carried a pavise.
n.
A course of notes so running into each other as to form one series or chain, to be sung in one breath to one syllable.
n.
One of the larger districts into which a country is divided for administering military affairs.
n.
The separation of a genus into its constituent species.
n.
One of the groups into which a fleet is divided.
n.
One who devises, or gives real estate by will; a testator; -- correlative to devisee.
n.
The portion separated by the divining of a mass or body; a distinct segment or section.
n.
The act or process of diving anything into parts, or the state of being so divided; separation.
n.
A grade or rank in classification; a portion of a tribe or of a class; or, in some recent authorities, equivalent to a subkingdom.
n.
Disunion; difference in opinion or feeling; discord; variance; alienation.
n.
Two companies of infantry maneuvering as one subdivision of a battalion.
n.
Difference of condition; state of distinction; distinction; contrast.
n.
One who, or that which, causes division.
n.
Two or more brigades under the command of a general officer.