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Graph of the Kempner function In number theory , the Kempner function S ( n ) {\displaystyle S(n)} [ 1 ] is defined for a given positive integer n {\displaystyle n} to be the smallest number s {\displaystyle s} such that n {\displaystyle n} divides the factorial s ! {\displaystyle s!} .
In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x). [1]
The smallest i such that a i < a 0 is called the stopping time of n. Similarly, the smallest k such that a k = 1 is called the total stopping time of n. [2] If one of the indexes i or k doesn't exist, we say that the stopping time or the total stopping time, respectively, is infinite. The Collatz conjecture asserts that the total stopping time ...
Then the cubicity of , denoted by (), is the smallest integer such that can be realized as an intersection graph of axis-parallel unit cubes in -dimensional Euclidean space. [ 2 ] The cubicity of a graph is closely related to the boxicity of a graph, denoted box ( G ) {\displaystyle \operatorname {box} (G)} .
Carmichael λ function: λ(n) for 1 ≤ n ≤ 1000 (compared to Euler φ function) In number theory, a branch of mathematics, the Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that holds for every integer a coprime to n.
Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients. Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer. Constant function: polynomial of degree zero, graph is a horizontal straight line
Graph of the fractional part of real numbers The fractional part or decimal part [ 1 ] of a non‐negative real number x {\displaystyle x} is the excess beyond that number's integer part . The latter is defined as the largest integer not greater than x , called floor of x or ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } .
If n is the smallest nonnegative integer, such that for some i, j, the element (i, j) of A n is positive, then n is the distance between vertex i and vertex j. A great example of how this is useful is in counting the number of triangles in an undirected graph G , which is exactly the trace of A 3 divided by 3 or 6 depending on whether the graph ...