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Divisor function σ 0 (n) up to n = 250 Sigma function σ 1 (n) up to n = 250 Sum of the squares of divisors, σ 2 (n), up to n = 250 Sum of cubes of divisors, σ 3 (n) up to n = 250. In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer.
A divisor of that is not a trivial divisor is known as a non-trivial divisor (or strict divisor [6]). A nonzero integer with at least one non-trivial divisor is known as a composite number , while the units −1 and 1 and prime numbers have no non-trivial divisors.
A divisor of an integer n is an integer m, for which n/m is again an integer (which is 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 so is −m. The tables below only list positive divisors.
One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common divisor. The common divisors can be found by dividing both numbers by successive integers from 2 to the smaller number b .
Divisor function d(n) up to n = 250 Prime-power factors In number theory , a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors . Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power.
The first thousand values of φ(n).The points on the top line represent φ(p) when p is a prime number, which is p − 1. [1]In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n.
Euclidean division is the mathematical formulation of the outcome of the usual process of division of integers. It asserts that, given two integers, a, the dividend, and b, the divisor, such that b ≠ 0, there are unique integers q, the quotient, and r, the remainder, such that a = bq + r and 0 ≤ r < | b |, where | b | denotes the absolute ...
An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n. Arithmetic functions are often extremely irregular (see table ), but some of them have series expansions in terms of Ramanujan's sum .