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For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and () is even; for a square integer, one divisor (namely ) is not paired with a distinct divisor and () is odd. Similarly, the number σ 1 ( n ) {\displaystyle \sigma _{1}(n)} is odd if and only if n is a square or twice a square.
In the SVG file, hover over a bar to see its statistics. The tables below list all of the divisors of the numbers 1 to 1000. A divisor of an integer n is an integer m, for which n/m is again an integer (which is necessarily also a divisor of n). For example, 3 is a divisor of 21, since 21/7 = 3 (and therefore 7 is also a divisor of 21).
Divisor function d(n) up to n = 250 Prime-power factors. In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power.
A classical example of this phenomenon [9] is given by the divisor summatory function, the summation function of d(n), the number of divisors of n: = = + + + + + + = An average order of an arithmetic function is some simpler or better-understood function which has the same summation function asymptotically, and ...
σ k (n) is the divisor function (i.e. the sum of the k-th powers of the divisors of n, including 1 and n). σ 0 (n), the number of divisors of n, is usually written d(n) and σ 1 (n), the sum of the divisors of n, is usually written σ(n). If s > 0,
The choice of the test depends on many properties of the research question. The vast majority of studies can be addressed by 30 of the 100 or so statistical tests in use . [ 3 ] [ 4 ] [ 5 ]
Fibonacci, in his 1202 book Liber Abaci [2] lists several methods for finding Egyptian fraction representations of a rational number. Of these, the first is to test whether the number is itself already a unit fraction, but the second is to search for a representation of the numerator as a sum of divisors of the denominator, as described above.
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: