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d() is the number of positive divisors of n, including 1 and n itself; σ() is the sum of the positive divisors of n, including 1 and n itselfs() is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n
For example, 6 is highly composite because d(6)=4 and d(n)=1,2,2,3,2 for n=1,2,3,4,5 respectively. A related concept is that of a largely composite number , a positive integer that has at least as many divisors as all smaller positive integers.
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.
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 divisors of n are all products of some or all prime factors of n (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then multiplying them. Divisors and properties related to divisors are shown in table of divisors.
The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10 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 be multiplied by some integer to produce n . {\displaystyle n.} [ 1 ] In this case, one also says that n {\displaystyle n ...
Therefore, 12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can thus be divided into a grid of 12-by-12 squares, with two squares along one edge (24/12 = 2) and five squares along the other (60/12 = 5).
This is true in the case of 6; 6's divisors are 1,2,3, and 6, but an abundant number is defined to be one where the sum of the divisors, excluding itself, is greater than the number itself; 1+2+3=6, so this condition is not met (and 6 is instead a perfect number). However all colossally abundant numbers are also superabundant numbers. [12]