floornk

FloorNk is a notation occasionally used in mathematical writing to denote the floor of the product Nk, i.e., the greatest integer not greater than the product of N and k. The floor function, denoted ⌊x⌋, maps a real number x to the largest integer less than or equal to x; floorNk is therefore defined as ⌊N k⌋. In typical use, N and k are real numbers, but they may also be integers depending on the problem setting.

Examples illustrate the idea: if N = 3 and k = 4, floorNk = ⌊12⌋ = 12. If N = 3.2 and k = 2, floorNk = ⌊6.4⌋ = 6. If N and k are integers, floorNk equals Nk. The quantity floorNk thus behaves as a step function in either variable, changing value only when the product Nk crosses an integer threshold.

Applications of floorNk arise in contexts where a continuous scaling is converted to a discrete count. This includes lattice-point counting in dilated regions, rounding decisions in algorithms that depend on the product of two parameters, and problems in number theory where the integer part of a scaled quantity is relevant. Because the floor operation is non-linear, floorNk can exhibit nontrivial behavior as N or k vary, even when Nk changes smoothly.

Notation notes: floorNk is not a universally fixed standard; some authors write ⌊Nk⌋ or floor(Nk), and others may define the expression differently in specialized contexts. When encountering floorNk, it is helpful to check the source for the exact definition and scope.

See also: floor function, lattice point counting, Beatty sequences.