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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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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
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
{\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
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
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
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
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
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
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
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
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
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
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
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
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
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
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. To see this, take
Semiprime
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
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
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
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
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, and
Abundant_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
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
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
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
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
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 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)
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
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
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
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
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
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
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
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
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
Integer whose multiples are digit rotations
(e.g. 9 in decimal) and the sum of the remainder is a multiple of the divisor. (This follows from the previous point.) Using the above technique, cyclic
Cyclic_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
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
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
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
Type of composite integer
{\displaystyle n} be a natural number. For base b > 1 {\displaystyle b>1} , let the function F b ( n ) {\displaystyle F_{b}(n)} be the digit sum of n {\displaystyle
Smith_number
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
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
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
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
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
Ten raised to an integer power
Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally
Power_of_10
Integers occurring in the coefficients of the Taylor series of 1/cosh t
Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically
Euler_numbers
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
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
Result of multiplying four instances of a number together
Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally
Fourth_power
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
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
Number that is more than the sum of its proper divisors
integer n for which the sum of divisors of n is less than 2n. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than
Deficient_number
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
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
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)
Formula whose values are the prime numbers
(d)}{2^{d}-1}}} , μ {\displaystyle \mu } is the Möbius function and d {\displaystyle d} runs through all divisors of p n # {\displaystyle p_{n}\#} , the primorial
Formula_for_primes
Fixed number that has received a name
the growth rate of the divisor function. It has relations to the gamma function and its derivatives as well as the zeta function and there exist many different
Mathematical_constant
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
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
Two raised to an integer power
Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally
Power_of_two
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
Numbers whose sum of divisors is twice the number minus 1
such that the sum of all divisors of n (the sum-of-divisors function σ(n)) is equal to 2n − 1, the sum of all proper divisors of n, s(n) = σ(n) − n, then
Almost_perfect_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
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
Type of positive integer
for each integer i between 0 and k, at least one of the greatest common divisors gcd(a, a + i) or gcd(a + i, a + k) is greater than 1. 16 is an Erdős–Woods
Erdős–Woods_number
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
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
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
Numbers k where x - phi(x) = k has many solutions
{\displaystyle k} and above 1. Here, ϕ {\displaystyle \phi } is Euler's totient function. There are infinitely many solutions to the equation for k {\displaystyle
Highly_cototient_number
Prime number of the form 2^n – 1
the former congruence must be true and 2p + 1 divides Mp. All composite divisors of prime-exponent Mersenne numbers are strong pseudoprimes to the base
Mersenne_prime
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
Composite number in number theory
number if and only if n {\displaystyle n} is square-free, and for all prime divisors p {\displaystyle p} of n {\displaystyle n} , it is true that p − 1
Carmichael_number
Type of composite number with an even number of digits
Somer–Lucas pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally
Vampire_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
DIVISOR FUNCTION
DIVISOR FUNCTION
Boy/Male
French, German
Honest Advisor; Brave Advisor
Boy/Male
Muslim
Divider.
Boy/Male
Indian
Divider
Boy/Male
Hindu, Indian
Beautiful Sky
Boy/Male
Indian
Advisor
Girl/Female
Arabic, Muslim, Sindhi
Advisor
Girl/Female
Biblical
Division.
Boy/Male
Muslim
Advisor.
Boy/Male
Muslim
Advisor
Boy/Male
Indian
Advisor
Girl/Female
Biblical
Division.
Girl/Female
Hindu, Indian, Tamil
Devine Wish
Boy/Male
Muslim
Advisor.
Boy/Male
American, Australian, British, English
Son of David; David's Son; Surname
Boy/Male
English
David's son. Surname.
Boy/Male
Muslim
Advisor
Girl/Female
Muslim
Advisor
Boy/Male
Muslim
Advisor
Girl/Female
Indian
Goddess Durga, Chief of the Goddess, Devee
Boy/Male
Indian
Advisor
DIVISOR FUNCTION
DIVISOR FUNCTION
Boy/Male
Hindu, Indian, Traditional
Lotus Feet of Radharani
Girl/Female
Australian, Greek, Hebrew
Gift of God; Gift; Well Loved
Boy/Male
Hindu
Lord Krishna
Boy/Male
Greek
Revered.
Girl/Female
Muslim/Islamic
Princess
Male
Chinese
studious.
Boy/Male
Anglo, British, Dutch, English, French, German, Hebrew, Swedish
Father in Rejoicing
Girl/Female
Muslim
Light
Girl/Female
Hebrew
Lord's vineyard.
Boy/Male
American, Anglo, Australian, British, English, French, Greek
Lives Near the Church; Ecclesiastical Locality
DIVISOR FUNCTION
DIVISOR FUNCTION
DIVISOR FUNCTION
DIVISOR FUNCTION
DIVISOR FUNCTION
n.
Difference of condition; state of distinction; distinction; contrast.
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.
That which divides or keeps apart; a partition.
n.
A conjecture; a guesser; one who makes out occult things.
n.
The act or process of diving anything into parts, or the state of being so divided; separation.
n.
One who devises, or gives real estate by will; a testator; -- correlative to devisee.
n.
A soldier who carried a pavise.
n.
The portion separated by the divining of a mass or body; a distinct segment or section.
n.
The number by which the dividend is divided.
n.
Two companies of infantry maneuvering as one subdivision of a battalion.
n.
The distribution of a discourse into parts; a part so distinguished.
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.
One of the groups into which a fleet is divided.
n.
One who, or that which, causes division.
n.
Separation of the members of a deliberative body, esp. of the Houses of Parliament, to ascertain the vote.
n.
The separation of a genus into its constituent species.
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.
One of the larger districts into which a country is divided for administering military affairs.
n.
Two or more brigades under the command of a general officer.
n.
Disunion; difference in opinion or feeling; discord; variance; alienation.