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...

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...

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...

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...

divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors...

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...

prime-counting functions. This article provides links to functions of both classes. An example of an arithmetic function is the divisor function whose value...

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...

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....

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...

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 {\displaystyle {\frac {b}{a}}}...

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...

called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal...

and chronic traumatic encephalopathy Divisor function in number theory, also denoted d or σ0 Ramanujan tau function Golden ratio (1.618...), although φ...

sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. The first few...

positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or...

{\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...

coefficients of the Ramanujan modular form Divisor function, an arithmetic function giving the number of divisors of an integer This disambiguation page lists...

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...

\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...

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} ...

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...

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...

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...

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...

its divisors (the sum-of-divisors function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding...

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...

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...

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...

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...