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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: A000203). The aliquot sum s ( n ) of n is the sum of the proper divisors (that is, the divisors excluding n itself, OEIS : A001065 ), and equals σ 1 ( n ) − n ; the aliquot ...
The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ 1 or the aliquot sum function s in the following way: [1] = = = > = = = If the s n-1 = 0 condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are convergent, the limit of these ...
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function . The various studies of the behaviour of the divisor function are sometimes called divisor problems .
The sum of proper divisors of a number 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; in symbols, () = where is the sum-of-divisors function.
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.
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or equivalently the Dirichlet convolution of an arithmetic function () with one:
In mathematics, the 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). The smallest pair of amicable numbers is ...
In mathematics, a superabundant number is a certain kind of natural number.A natural number n is called superabundant precisely when, for all m < n: < ()where σ denotes the sum-of-divisors function (i.e., the sum of all positive divisors of n, including n itself).