Concept
Abundant number
Related concepts (7)
Deficient number
In number theory, a deficient number or defective number is a positive integer n for which the sum of divisors of n is less than 2n. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than n. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient. Denoting by σ(n) the sum of divisors, the value 2n − σ(n) is called the number's deficiency. In terms of the aliquot sum s(n), the deficiency is n − s(n).
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.
Aliquot sum
In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself. That is, It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number. For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are 1, 2, 3, 4, and 6, so the aliquot sum of 12 is 16 i.
Semiperfect number
In number theory, 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. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. The first few semiperfect numbers are: 6, 12, 18, 20, 24, 28, 30, 36, 40, ... Every multiple of a semiperfect number is semiperfect. A semiperfect number that is not divisible by any smaller semiperfect number is called primitive.
Weird number
In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself. The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but not weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2 + 4 + 6 = 12.
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, where is the sum-of-divisors function.
Divisor
In mathematics, a divisor of an integer , also called a factor of , is an integer that may be multiplied by some integer to produce . In this case, one also says that is a multiple of An integer is divisible or evenly divisible by another integer if is a divisor of ; this implies dividing by leaves no remainder. An integer n is divisible by a nonzero integer m if there exists an integer k such that . This is written as Other ways of saying the same thing are that m divides n, m is a divisor of n, m is a factor of n, and n is a multiple of m.