OsqrtNk

OsqrtNk is a mathematical notation used to denote a transform applied to a sequence of nonnegative integers indexed by k. In its standard form, OsqrtNk(N) refers to the sequence obtained by taking the integer part of the square root of each term: for a given sequence N = {N_k}, the OsqrtNk transform is the sequence { floor( sqrt( N_k ) ) } for k = 1, 2, 3, ... . The name combines the letter O to indicate an operator or transformation, sqrt for the square-root, and N_k as the k-th parameter of the sequence.

Properties and behavior

OsqrtNk preserves the pointwise order: if N_k ≤ M_k for all k, then floor( sqrt( N_k ) ) ≤ floor( sqrt( M_k ) ) for all k. The transform reduces the growth rate of a sequence; if N_k grows like a power of k, the transformed sequence grows roughly with the square root of that growth. The transform is not injective, since distinct N_k values can map to the same floor(sqrt()) value due to flooring.

Examples

If N_k = k^2, then OsqrtNk(N)_k = floor( sqrt( k^2 ) ) = k. If N_k = k, then OsqrtNk(N)_k = floor( sqrt( k ) ), which grows like the square root of k. If N_k is constant, the transformed sequence is constant as well.

Applications and scope

OsqrtNk is used in theoretical contexts to bound or approximate costs and quantities that scale with the square root of a parameter sequence. It appears in discussions of algorithmic analyses, combinatorial estimates, and sequence transformations where a square-root relation is natural. The term is not tied to a single canonical source and may appear with small variations in notation, such as Osqrt(N_k) or O_sqrt(N_k).

See also

floor function, square root, sequence transformation, asymptotic analysis

Notes

This article documents a conventional definition used in teaching and theoretical discussions; specific terminology may vary across sources.