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Oct 19, 2023 · A number n is said to be an Abundant Number if the sum of all the proper divisors of the number denoted by sum (n) is greater than the value of the number n. And the difference between these two values is called abundance.
An abundant number is a natural number n for which the sum of divisors σ(n) satisfies σ(n) > 2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n) satisfies s(n) > n. The abundance of a natural number is the integer σ(n) − 2n (equivalently, s(n) − n).
An abundant number (also known as excessive numbers) is a positive integer such that the sum of its proper divisors is greater than the number itself. Or equivalently, a positive integer n n is said to be abundant if \sigma_1 (n) > 2n σ1(n)> 2n, where \sigma_1 (n) σ1(n) denotes the sum of factors of n n.
An abundant number, sometimes also called an excessive number, is a positive integer n for which s (n)=sigma (n)-n>n, (1) where sigma (n) is the divisor function and s (n) is the restricted divisor function. The quantity sigma (n)-2n is sometimes called the abundance.
Aug 8, 2023 · In number theory, “an abundant number is a positive integer that is smaller than the sum of its proper divisors. The proper divisors of a number are all its positive divisors excluding itself”. In other words, “an abundant number n is one for which the sum of its proper divisors is greater than n ”. Examples:
Abundant numbers are positive integers for which the sum of their proper divisors exceeds the number itself. This means that when you add up all the factors of a number, excluding the number, the total is greater than the number.
Aug 22, 2024 · The abundancy of a number n is defined as the ratio sigma (n)/n, where sigma (n) is the divisor function. For n=1, 2, ..., the first few values are 1, 3/2, 4/3, 7/4, 6/5, 2, 8/7, 15/8, 13/9, 9/5, 12/11, 7/3, 14/13, ...
Multiplying a perfect number by some integer x x gives an abundant number (as long as x>1 x> 1). Given a pair of amicable numbers, the lesser of the two is abundant, its proper divisors adding up to the greater of the two.
Feb 15, 2024 · History. Abundant number. Let $\sigma ( n )$ denote the sum of the distinct divisors of an integer $n$ (cf. Divisor; Number of divisors). The integer $n$ is called abundant if $\sigma ( n ) > 2 n$; deficient if $\sigma ( n ) < 2 n$; and perfect if $\sigma ( n ) = 2 n$ (cf. also Perfect number).
Abundant numbers are positive integers for which the sum of their proper divisors (excluding the number itself) is greater than the number. This characteristic places abundant numbers in a unique category within the study of integers, where they contrast with perfect and deficient numbers, revealing interesting properties regarding their ...
