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In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. Here σ(n) is the divisor function: the sum of all positive divisors of n, including n itself. The value σ(n) − 2n is called the abundance of n. An equivalent definition is that the proper divisors of the number (the divisors except the number itself) sum to more than the number. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
In mathematics the divisor function σa(n) is defined as the sum of the ath powers of the divisors of n, or The notations d(n) and (the tau function) are also used to denote σ0(n), or the number of divisors of n. ...
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...
The first few abundant numbers (sequence A005101 in OEIS) are: The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...
- 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, …
As an example, consider the number 24. Its divisors are 1, 2, 3, 4, 6, 8, 12 and 24, whose sum is 60. Because 60 is more than 2 × 24, the number 24 is abundant. Its abundance is 60 − 2 × 24 = 12.
The smallest odd abundant number is 945. Marc Deléglise showed in 1998 that the natural density of abundant numbers is between 0.2474 and 0.2480. 1998(MCMXCVIII) is a common year starting on Thursday of the Gregorian calendar, and was designated the International Year of the Ocean. ...
In mathematics, a sequence a1, a2, ... , an, with the aj positive integers and aj < aj+1 for all j, has natural density α, where 0 ≤ α ≤ 1, if the proportion of natural numbers included as some aj is asymptotic to α. More formally, if we define the counting function A(x) as the...
Infinitely many even and odd abundant numbers exist. Every proper multiple of a perfect number, and every multiple of an abundant number is abundant. Also, every integer greater than 20161 can be written as the sum of two abundant numbers. An abundant number which is not a semiperfect number is called a weird number; an abundant number with abundance 1 is called a quasiperfect number. In mathematics, any integer is either even or odd. ...
In math, a perfect square is defined as an integer which is the sum of its proper positive divisors, excluding itself. ...
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
In mathematics, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. ...
In mathematics, a weird number is a natural number that is abundant but not semiperfect. ...
In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. ...
Closely related to abundant numbers are perfect numbers with σ(n) = 2n, and deficient numbers with σ(n) < 2n. The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica (circa 100). In math, a perfect square is defined as an integer which is the sum of its proper positive divisors, excluding itself. ...
In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. ...
Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), and they can be used for ordering (this is...
Nicomachus (c. ...
-1...
See also
A highly composite number is a positive integer which has more divisors than any positive integer below it. ...
In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. ...
In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. ...
External links
References - M. Deléglise, "Bounds for the density of abundant integers," Experimental Math., 7:2 (1998) p. 137-143.
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