Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, s(a)=b and s(b)=a, where s(n)=σ(n)-n is equal to the sum of positive divisors of n except n itself (see also divisor function).
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, and 1 + 2 + 3 = 6, so 6 is a perfect number.
In mathematics, a multiply perfect number is a generalization of a perfect number.
In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example.
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 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.
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. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far.
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 of the number is greater than the number, but no subset of those divisors sums to the number itself.
In mathematics, an almost perfect number (sometimes also called slightly defective or least deficientnumber) is a natural number n such that the sum of all divisors of n (the sum-of-divisors function σ(n)) is equal to 2n − 1, the sum of all proper divisors of n, s(n) = σ(n) − n, then being equal to n − 1. The only known almost perfect numbers are powers of 2 with non-negative exponents (sequence A000079 in the OEIS). Therefore the only known odd almost perfect number is 20 = 1, and the only known even almost perfect numbers are those of the form 2k for some positive integer k; however, it has not been shown that all almost perfect numbers are of this form. It is known that an odd almost perfect number greater than 1 would have at least six prime factors.
In mathematics, a k-hyperperfect number is a natural number n for which the equality n = 1 + k(σ(n) − n − 1) holds, where σ(n) is the divisor function (i.e., the sum of all positive divisors of n). A hyperperfect number is a k-hyperperfect number for some integer k. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.
In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it.
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. 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.
In mathematics, a harmonic divisor number, or Ore number, is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are:
In mathematics, an untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function. Their study goes back at least to Abu Mansur al-Baghdadi, who observed that both 2 and 5 are untouchable.
In number theory, a practical number or panarithmic number is a positive integer such that all smaller positive integers can be represented as sums of distinct divisors of . For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2.
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms.
In number theory, a colossally abundant number is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio is a colossally abundant number. It is a stronger restriction than that of a superabundant number, but not strictly stronger than that of an abundant number.
In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly n-tuple.
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite. The impolite numbers are exactly the powers of two, and the polite numbers are the natural numbers that are not powers of two.
In mathematics, a natural number a is a unitary divisor of a number b if a is a divisor of b and if a and are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number.
In number theory, a Descartes number is an odd number which would have been an odd perfect number if one of its composite factors were prime. They are named after René Descartes who observed that the number D = 32⋅72⋅112⋅132⋅22021 = (3⋅1001)2 ⋅ (22⋅1001 − 1) = 198585576189 would be an odd perfect number if only 22021 were a prime number, since the sum-of-divisors function for D would satisfy, if 22021 were prime,
