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a highly abundant number has a sum of positive divisors that is greater than any lesser number; that is, σ(n) > σ(m) for every positive integer m < n. Counterintuitively, the first seven highly abundant numbers are not abundant numbers. a prime number has only 1 and itself as divisors; that is, d(n) = 2
Demonstration, with Cuisenaire rods, of the first four highly composite numbers: 1, 2, 4, 6. A highly composite number is 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 integer N is highly composite if d(N) > d(n) for all n < N.
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]
The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ 0 (n), or the number-of-divisors function [1] [2] (OEIS: A000005). When z is 1, the function is called the sigma function or sum-of-divisors function , [ 1 ] [ 3 ] and the subscript is often omitted, so σ ( n ) is the same as σ 1 ( n ) ( OEIS ...
An abundant number whose abundance is greater than any lower number is called a highly abundant number, and one whose relative abundance (i.e. s(n)/n ) is greater than any lower number is called a superabundant number; Every integer greater than 20161 can be written as the sum of two abundant numbers. The largest even number that is not the sum ...
The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A002201 in the OEIS) are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither ...
An extravagant number has fewer digits than its prime factorization. The first in decimal: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30 (sequence A046760 in the OEIS). An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital.
More generally, an a-by-b rectangle can be covered with square tiles of side length c only if c is a common divisor of a and b. For example, a 24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares.