Definition
The sum of positive divisors function σx(n), for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n. It can be expressed in sigma notation as
where is shorthand for "d divides n". The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ0(n), or the number-of-divisors function (sequence A000005 in OEIS). When x is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted, so σ(n) is equivalent to σ1(n) ( A000203).
The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, A001065), and equals σ1(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.
Read more about this topic: Divisor Function
Famous quotes containing the word definition:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)